Drorbn - User contributions [en]

06-1350/Syzygies in Asymptote

2008-12-01T16:28:59Z

Andy: /* Installation */

{{06-1350/Navigation}}

For a condensed version of this page, check [[06-1350/Syzygies in Asymptote in Brief]].

===Disclaimer===
These instructions (and the program they describe) are a work in progress and should be considered highly unreliable.

===Installation===
To use the syzygy script, you need to have a working version of Asymptote. Asymptote is installed on coxeter. To install on another computer, see [http://asymptote.sourceforge.net Asymptote]. Instructions for installing the program on several OSes is given in the documentation at the Asymptote website. The documentation also gives helpful instructions on how to run a script in Asymptote to produce a picture. You should also have (or install) a variant of TeX on your system, such as MiKTeX, so that Asymptote can typeset labels. The <tt>syzygy</tt> package is now part of the standard Asymptote distribution.

===Braids===
Once installed, we can draw a braid in Asymptote:

<pre>
import syzygy; // Accesses the syzygy module.
Braid b; // Start a new braid.
b.n=3; // The braid has three strands.
// The strands are numbered left to right starting at 0.
b.add(bp,0); // Add a overcrossing component starting at strand 0,
// the leftmost strand.
b.add(bm,1); // Add an undercrossing starting at strand 1.
b.add(phi,0); // Add a trivalent vertex that merges strands 0 and 1.
// Strand 2 is now renumbered as strand 1.
b.draw(); // Draw the resulting braid.
</pre>

When saved into an asy file, say <code>mybraid.asy</code> and run with Asymptote, the result is a picture:

[[Image:06-1350-mybraid.png|center]]

===Relations===
====Drawing====
To define a relation, we first define two braids, and then stick them into a <code>Relation</code> structure. The below script generates an R3 relation.

<pre>
import syzygy; // Access the syzygy module.
Braid l; // Define the left hand side of the relation.
l.n=3; l.add(bp,0); l.add(bp,1); l.add(bp,0);
Braid r; // Define the right hand side of the relation.
r.n=3; r.add(bp,1); r.add(bp,0); r.add(bp,1);

Relation r3; // Define a relation.
r3.lsym="\rho_3"; // Give the relation a name for when it is written in functional form.
r3.codename="rho3"; // Give the relation a name to be used by Mathematica.
r3.lhs=l; r3.rhs=r;
r3.draw();
</pre>

When saved into an asy file and run, this draws the two sides of the relation. If TeX is installed, Asymptote will also put a lovely equals sign, typeset by TeX, between the two figures.

[[Image:06-1350-R3-asy.png|center]]

====Outputting Equations====
We can also get useful equations out of the relation. The method <code>r3.toFormula()</code> will produce a string that is the formula for the relation.

<pre>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</pre>

This string can be written out to the standard output by <code>write(r3.toFormula())</code>. It can be written to a file by <code>file f=output("filename.txt"); write(f, r3.toFormula())</code>. The string is formatted so it can be put into TeX or a wiki page using math mode:

<math>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</math>

The method <code>r3.toLinear()</code> produces the formula in linear form:

<math>
\rho_3(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4) - b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3)
</math>

and <code>r3.toCode()</code> produces a version of the relation that can be used in Mathematica:

<pre>
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] + bp[x1, x3, x4]
- bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]
</pre>

A few relations, such as <code>r3</code>, are already defined in <code>syzygy.asy</code> but more should be added.

====Applying====
Now that we have relations, we can apply them to bigger braids. Let's start with the braid in the <math>\Phi</math> around B syzygy:

<pre>
import syzygy;
Braid b;
b.n=4;
b.add(bp,2);
b.add(bp,0);
b.add(bp,1);
b.add(bp,0);
b.add(bp,2);
b.add(phi,1);
</pre>

[[Image:06-1350-pbstart.png|center]]

After skipping the lowest knot, we can apply R3 to the next three knots:

<pre>
Braid bb=apply(r3, b, 1, 0);
</pre>

here <code>apply(r, b, k, n)</code> means we are applying the relation <code>r</code> to the braid <code>b</code> at the place in the braid found by counting <code>k</code> components up from the bottom component and <code>n</code> strands in from the leftmost strand. <code>apply</code> does not modify the original braid, but returns the result of applying the relation (stored here as <code>bb</code>):

[[Image:06-1350-pbnext.png|center]]

This went from the left hand side of the relation to the right hand side. To apply a relation in reverse, simply prefix it by a minus sign. For example <code>apply(-r3, bb, 1, 0)</code> will yield a braid equivalent to our original. When applying a relation, the script first checks that the one side of the relation matches that portion of the braid, and will give a (somewhat cryptic) error if the relation cannot be applied.

In our braids, the components are placed from bottom to top in a fixed order. Sometimes when building syzygies, it is neccessary to swap the order that these components occur. This is done by the <code>swap</code> method. For instance, starting from <code>b</code>, we can swap the two bottom crossings:

<pre>
Braid swapped=b.swap(0,1);
</pre>

[[Image:06-1350-pbswap.png|center]]

Remember that components are ordered from bottom to top, starting at 0. Again, the script checks to make sure the swap is valid (ie. changing the order of the two components, doesn't actually change the knot) and will issue an error if it isn't.

===Syzygies===
One could manually apply relations and swaps, and make a whole bunch of braids, but it would be annoying to keep track of them all. Thankfully, the <code>Syzygy</code> structure does that for us. For example, here is the complete code for the <math>\Phi</math> around B syzygy:

<pre>
import syzygy;

// Phi around B
Braid initial;
initial.n=4;
initial.add(bp,2);
initial.add(bp,0);
initial.add(bp,1);
initial.add(bp,0);
initial.add(bp,2);
initial.add(phi,1);

Syzygy pb;
pb.lsym="\Phi B";
pb.codename="PhiAroundB";
pb.initial=initial;
pb.apply(r3,1,0);
pb.apply(r4a,3,1);
pb.swap(2,3);
pb.apply(r4b,0,1);
pb.apply(-r3,1,0);
pb.apply(-r4a,0,0);
pb.swap(2,3);
pb.apply(-r4b,3,0);
pb.apply(r3,1,1);

pb.draw();
</pre>

and the result

[[Image:06-1350-PhiAroundB.png|center]]

Again, like relations, we can use <code>pb.toLinear()</code>

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

and <code>pb.toCode()</code>

<pre>
PhiAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + rho4a[x1 + x5, x2, x3, x4]
+ rho4b[x1 + x2, x3, x4, x5] - rho3[x1, x2, x3 + x4, x5] - rho4a[x1, x2, x3, x4]
- rho4b[x1, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5]
</pre>

to give the formulas for the syzygies.

The <code>Syzygy</code> structure assumes that after the last application of a relation, the braid is in the same form as the start, so it won't draw the last braid. This is annoying when building a syzygy, so it can be turned off by <code>pb.cyclic=false;</code> If you set <code>bp.showall=true;</code> the syzygy will draw all changes to the braid, including swaps. Finally, setting <code>bp.number=true;</code> will print numbers on the diagrams so you can follow them around.

===Conclusion===
Example syzygyies and the latest version of the script can be found in the syzygy directory on the course Subversion repository [http://katlas.math.toronto.edu/svn/06-1350/syzygy/trunk/syzygy/]. Please contact me if you have any questions or suggestions. Good luck and happy syzyging!

06-1350/Syzygies in Asymptote

2006-12-09T17:53:24Z

Andy: /* Conclusion */

{{06-1350/Navigation}}

For a condensed version of this page, check [[06-1350/Syzygies in Asymptote in Brief]].

===Disclaimer===
These instructions (and the program they describe) are a work in progress and should be considered highly unreliable.

===Installation===
To use the syzygy script, first install [http://asymptote.sourceforge.net Asymptote]. Instructions for installing the program on several OSes is given in the documentation at the Asymptote website. The documentation also gives helpful instructions on how to run a script in Asymptote to produce a picture. Once installed, download [http://katlas.math.toronto.edu/svn/06-1350/syzygy/trunk/syzygy/syzygy.asy syzygy.asy] and put it in a directory where Asymptote can find it. You should also have (or install) a variant of TeX on your system, such as MiKTeX, so that Asymptote can typeset labels.

===Braids===
Once installed, we can draw a braid in Asymptote:

<pre>
import syzygy; // Accesses the syzygy module.
Braid b; // Start a new braid.
b.n=3; // The braid has three strands.
// The strands are numbered left to right starting at 0.
b.add(bp,0); // Add a overcrossing component starting at strand 0,
// the leftmost strand.
b.add(bm,1); // Add an undercrossing starting at strand 1.
b.add(phi,0); // Add a trivalent vertex that merges strands 0 and 1.
// Strand 2 is now renumbered as strand 1.
b.draw(); // Draw the resulting braid.
</pre>

When saved into an asy file, say <code>mybraid.asy</code> and run with Asymptote, the result is a picture:

[[Image:06-1350-mybraid.png|center]]

===Relations===
====Drawing====
To define a relation, we first define two braids, and then stick them into a <code>Relation</code> structure. The below script generates an R3 relation.

<pre>
import syzygy; // Access the syzygy module.
Braid l; // Define the left hand side of the relation.
l.n=3; l.add(bp,0); l.add(bp,1); l.add(bp,0);
Braid r; // Define the right hand side of the relation.
r.n=3; r.add(bp,1); r.add(bp,0); r.add(bp,1);

Relation r3; // Define a relation.
r3.lsym="\rho_3"; // Give the relation a name for when it is written in functional form.
r3.codename="rho3"; // Give the relation a name to be used by Mathematica.
r3.lhs=l; r3.rhs=r;
r3.draw();
</pre>

When saved into an asy file and run, this draws the two sides of the relation. If TeX is installed, Asymptote will also put a lovely equals sign, typeset by TeX, between the two figures.

[[Image:06-1350-R3-asy.png|center]]

====Outputting Equations====
We can also get useful equations out of the relation. The method <code>r3.toFormula()</code> will produce a string that is the formula for the relation.

<pre>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</pre>

This string can be written out to the standard output by <code>write(r3.toFormula())</code>. It can be written to a file by <code>file f=output("filename.txt"); write(f, r3.toFormula())</code>. The string is formatted so it can be put into TeX or a wiki page using math mode:

<math>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</math>

The method <code>r3.toLinear()</code> produces the formula in linear form:

<math>
\rho_3(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4) - b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3)
</math>

and <code>r3.toCode()</code> produces a version of the relation that can be used in Mathematica:

<pre>
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] + bp[x1, x3, x4]
- bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]
</pre>

A few relations, such as <code>r3</code>, are already defined in <code>syzygy.asy</code> but more should be added.

====Applying====
Now that we have relations, we can apply them to bigger braids. Let's start with the braid in the <math>\Phi</math> around B syzygy:

<pre>
import syzygy;
Braid b;
b.n=4;
b.add(bp,2);
b.add(bp,0);
b.add(bp,1);
b.add(bp,0);
b.add(bp,2);
b.add(phi,1);
</pre>

[[Image:06-1350-pbstart.png|center]]

After skipping the lowest knot, we can apply R3 to the next three knots:

<pre>
Braid bb=apply(r3, b, 1, 0);
</pre>

here <code>apply(r, b, k, n)</code> means we are applying the relation <code>r</code> to the braid <code>b</code> at the place in the braid found by counting <code>k</code> components up from the bottom component and <code>n</code> strands in from the leftmost strand. <code>apply</code> does not modify the original braid, but returns the result of applying the relation (stored here as <code>bb</code>):

[[Image:06-1350-pbnext.png|center]]

This went from the left hand side of the relation to the right hand side. To apply a relation in reverse, simply prefix it by a minus sign. For example <code>apply(-r3, bb, 1, 0)</code> will yield a braid equivalent to our original. When applying a relation, the script first checks that the one side of the relation matches that portion of the braid, and will give a (somewhat cryptic) error if the relation cannot be applied.

In our braids, the components are placed from bottom to top in a fixed order. Sometimes when building syzygies, it is neccessary to swap the order that these components occur. This is done by the <code>swap</code> method. For instance, starting from <code>b</code>, we can swap the two bottom crossings:

<pre>
Braid swapped=b.swap(0,1);
</pre>

[[Image:06-1350-pbswap.png|center]]

Remember that components are ordered from bottom to top, starting at 0. Again, the script checks to make sure the swap is valid (ie. changing the order of the two components, doesn't actually change the knot) and will issue an error if it isn't.

===Syzygies===
One could manually apply relations and swaps, and make a whole bunch of braids, but it would be annoying to keep track of them all. Thankfully, the <code>Syzygy</code> structure does that for us. For example, here is the complete code for the <math>\Phi</math> around B syzygy:

<pre>
import syzygy;

// Phi around B
Braid initial;
initial.n=4;
initial.add(bp,2);
initial.add(bp,0);
initial.add(bp,1);
initial.add(bp,0);
initial.add(bp,2);
initial.add(phi,1);

Syzygy pb;
pb.lsym="\Phi B";
pb.codename="PhiAroundB";
pb.initial=initial;
pb.apply(r3,1,0);
pb.apply(r4a,3,1);
pb.swap(2,3);
pb.apply(r4b,0,1);
pb.apply(-r3,1,0);
pb.apply(-r4a,0,0);
pb.swap(2,3);
pb.apply(-r4b,3,0);
pb.apply(r3,1,1);

pb.draw();
</pre>

and the result

[[Image:06-1350-PhiAroundB.png|center]]

Again, like relations, we can use <code>pb.toLinear()</code>

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

and <code>pb.toCode()</code>

<pre>
PhiAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + rho4a[x1 + x5, x2, x3, x4]
+ rho4b[x1 + x2, x3, x4, x5] - rho3[x1, x2, x3 + x4, x5] - rho4a[x1, x2, x3, x4]
- rho4b[x1, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5]
</pre>

to give the formulas for the syzygies.

The <code>Syzygy</code> structure assumes that after the last application of a relation, the braid is in the same form as the start, so it won't draw the last braid. This is annoying when building a syzygy, so it can be turned off by <code>pb.cyclic=false;</code> If you set <code>bp.showall=true;</code> the syzygy will draw all changes to the braid, including swaps. Finally, setting <code>bp.number=true;</code> will print numbers on the diagrams so you can follow them around.

===Conclusion===
Example syzygyies and the latest version of the script can be found in the syzygy directory on the course Subversion repository [[http://katlas.math.toronto.edu/svn/06-1350/syzygy/trunk/syzygy/]]. Please contact me if you have any questions or suggestions. Good luck and happy syzyging!

06-1350/Syzygies in Asymptote

2006-12-09T17:53:02Z

Andy: Changed link to SVN.

{{06-1350/Navigation}}

For a condensed version of this page, check [[06-1350/Syzygies in Asymptote in Brief]].

===Disclaimer===
These instructions (and the program they describe) are a work in progress and should be considered highly unreliable.

===Installation===
To use the syzygy script, first install [http://asymptote.sourceforge.net Asymptote]. Instructions for installing the program on several OSes is given in the documentation at the Asymptote website. The documentation also gives helpful instructions on how to run a script in Asymptote to produce a picture. Once installed, download [http://katlas.math.toronto.edu/svn/06-1350/syzygy/trunk/syzygy/syzygy.asy syzygy.asy] and put it in a directory where Asymptote can find it. You should also have (or install) a variant of TeX on your system, such as MiKTeX, so that Asymptote can typeset labels.

===Braids===
Once installed, we can draw a braid in Asymptote:

<pre>
import syzygy; // Accesses the syzygy module.
Braid b; // Start a new braid.
b.n=3; // The braid has three strands.
// The strands are numbered left to right starting at 0.
b.add(bp,0); // Add a overcrossing component starting at strand 0,
// the leftmost strand.
b.add(bm,1); // Add an undercrossing starting at strand 1.
b.add(phi,0); // Add a trivalent vertex that merges strands 0 and 1.
// Strand 2 is now renumbered as strand 1.
b.draw(); // Draw the resulting braid.
</pre>

When saved into an asy file, say <code>mybraid.asy</code> and run with Asymptote, the result is a picture:

[[Image:06-1350-mybraid.png|center]]

===Relations===
====Drawing====
To define a relation, we first define two braids, and then stick them into a <code>Relation</code> structure. The below script generates an R3 relation.

<pre>
import syzygy; // Access the syzygy module.
Braid l; // Define the left hand side of the relation.
l.n=3; l.add(bp,0); l.add(bp,1); l.add(bp,0);
Braid r; // Define the right hand side of the relation.
r.n=3; r.add(bp,1); r.add(bp,0); r.add(bp,1);

Relation r3; // Define a relation.
r3.lsym="\rho_3"; // Give the relation a name for when it is written in functional form.
r3.codename="rho3"; // Give the relation a name to be used by Mathematica.
r3.lhs=l; r3.rhs=r;
r3.draw();
</pre>

When saved into an asy file and run, this draws the two sides of the relation. If TeX is installed, Asymptote will also put a lovely equals sign, typeset by TeX, between the two figures.

[[Image:06-1350-R3-asy.png|center]]

====Outputting Equations====
We can also get useful equations out of the relation. The method <code>r3.toFormula()</code> will produce a string that is the formula for the relation.

<pre>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</pre>

This string can be written out to the standard output by <code>write(r3.toFormula())</code>. It can be written to a file by <code>file f=output("filename.txt"); write(f, r3.toFormula())</code>. The string is formatted so it can be put into TeX or a wiki page using math mode:

<math>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</math>

The method <code>r3.toLinear()</code> produces the formula in linear form:

<math>
\rho_3(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4) - b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3)
</math>

and <code>r3.toCode()</code> produces a version of the relation that can be used in Mathematica:

<pre>
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] + bp[x1, x3, x4]
- bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]
</pre>

A few relations, such as <code>r3</code>, are already defined in <code>syzygy.asy</code> but more should be added.

====Applying====
Now that we have relations, we can apply them to bigger braids. Let's start with the braid in the <math>\Phi</math> around B syzygy:

<pre>
import syzygy;
Braid b;
b.n=4;
b.add(bp,2);
b.add(bp,0);
b.add(bp,1);
b.add(bp,0);
b.add(bp,2);
b.add(phi,1);
</pre>

[[Image:06-1350-pbstart.png|center]]

After skipping the lowest knot, we can apply R3 to the next three knots:

<pre>
Braid bb=apply(r3, b, 1, 0);
</pre>

here <code>apply(r, b, k, n)</code> means we are applying the relation <code>r</code> to the braid <code>b</code> at the place in the braid found by counting <code>k</code> components up from the bottom component and <code>n</code> strands in from the leftmost strand. <code>apply</code> does not modify the original braid, but returns the result of applying the relation (stored here as <code>bb</code>):

[[Image:06-1350-pbnext.png|center]]

This went from the left hand side of the relation to the right hand side. To apply a relation in reverse, simply prefix it by a minus sign. For example <code>apply(-r3, bb, 1, 0)</code> will yield a braid equivalent to our original. When applying a relation, the script first checks that the one side of the relation matches that portion of the braid, and will give a (somewhat cryptic) error if the relation cannot be applied.

In our braids, the components are placed from bottom to top in a fixed order. Sometimes when building syzygies, it is neccessary to swap the order that these components occur. This is done by the <code>swap</code> method. For instance, starting from <code>b</code>, we can swap the two bottom crossings:

<pre>
Braid swapped=b.swap(0,1);
</pre>

[[Image:06-1350-pbswap.png|center]]

Remember that components are ordered from bottom to top, starting at 0. Again, the script checks to make sure the swap is valid (ie. changing the order of the two components, doesn't actually change the knot) and will issue an error if it isn't.

===Syzygies===
One could manually apply relations and swaps, and make a whole bunch of braids, but it would be annoying to keep track of them all. Thankfully, the <code>Syzygy</code> structure does that for us. For example, here is the complete code for the <math>\Phi</math> around B syzygy:

<pre>
import syzygy;

// Phi around B
Braid initial;
initial.n=4;
initial.add(bp,2);
initial.add(bp,0);
initial.add(bp,1);
initial.add(bp,0);
initial.add(bp,2);
initial.add(phi,1);

Syzygy pb;
pb.lsym="\Phi B";
pb.codename="PhiAroundB";
pb.initial=initial;
pb.apply(r3,1,0);
pb.apply(r4a,3,1);
pb.swap(2,3);
pb.apply(r4b,0,1);
pb.apply(-r3,1,0);
pb.apply(-r4a,0,0);
pb.swap(2,3);
pb.apply(-r4b,3,0);
pb.apply(r3,1,1);

pb.draw();
</pre>

and the result

[[Image:06-1350-PhiAroundB.png|center]]

Again, like relations, we can use <code>pb.toLinear()</code>

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

and <code>pb.toCode()</code>

<pre>
PhiAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + rho4a[x1 + x5, x2, x3, x4]
+ rho4b[x1 + x2, x3, x4, x5] - rho3[x1, x2, x3 + x4, x5] - rho4a[x1, x2, x3, x4]
- rho4b[x1, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5]
</pre>

to give the formulas for the syzygies.

The <code>Syzygy</code> structure assumes that after the last application of a relation, the braid is in the same form as the start, so it won't draw the last braid. This is annoying when building a syzygy, so it can be turned off by <code>pb.cyclic=false;</code> If you set <code>bp.showall=true;</code> the syzygy will draw all changes to the braid, including swaps. Finally, setting <code>bp.number=true;</code> will print numbers on the diagrams so you can follow them around.

===Conclusion===
Example syzygyies and the last version of the script can be found in the syzygy directory on the course Subversion repository [[http://katlas.math.toronto.edu/svn/06-1350/syzygy/trunk/syzygy/]]. Please contact me if you have any questions or suggestions. Good luck and happy syzyging!

06-1350/Syzygies in Asymptote

2006-12-09T17:48:30Z

Andy: Changed syzygy.asy link to the SVN repository.

{{06-1350/Navigation}}

For a condensed version of this page, check [[06-1350/Syzygies in Asymptote in Brief]].

===Disclaimer===
These instructions (and the program they describe) are a work in progress and should be considered highly unreliable.

===Installation===
To use the syzygy script, first install [http://asymptote.sourceforge.net Asymptote]. Instructions for installing the program on several OSes is given in the documentation at the Asymptote website. The documentation also gives helpful instructions on how to run a script in Asymptote to produce a picture. Once installed, download [http://katlas.math.toronto.edu/svn/06-1350/syzygy/trunk/syzygy/syzygy.asy syzygy.asy] and put it in a directory where Asymptote can find it. You should also have (or install) a variant of TeX on your system, such as MiKTeX, so that Asymptote can typeset labels.

===Braids===
Once installed, we can draw a braid in Asymptote:

<pre>
import syzygy; // Accesses the syzygy module.
Braid b; // Start a new braid.
b.n=3; // The braid has three strands.
// The strands are numbered left to right starting at 0.
b.add(bp,0); // Add a overcrossing component starting at strand 0,
// the leftmost strand.
b.add(bm,1); // Add an undercrossing starting at strand 1.
b.add(phi,0); // Add a trivalent vertex that merges strands 0 and 1.
// Strand 2 is now renumbered as strand 1.
b.draw(); // Draw the resulting braid.
</pre>

When saved into an asy file, say <code>mybraid.asy</code> and run with Asymptote, the result is a picture:

[[Image:06-1350-mybraid.png|center]]

===Relations===
====Drawing====
To define a relation, we first define two braids, and then stick them into a <code>Relation</code> structure. The below script generates an R3 relation.

<pre>
import syzygy; // Access the syzygy module.
Braid l; // Define the left hand side of the relation.
l.n=3; l.add(bp,0); l.add(bp,1); l.add(bp,0);
Braid r; // Define the right hand side of the relation.
r.n=3; r.add(bp,1); r.add(bp,0); r.add(bp,1);

Relation r3; // Define a relation.
r3.lsym="\rho_3"; // Give the relation a name for when it is written in functional form.
r3.codename="rho3"; // Give the relation a name to be used by Mathematica.
r3.lhs=l; r3.rhs=r;
r3.draw();
</pre>

When saved into an asy file and run, this draws the two sides of the relation. If TeX is installed, Asymptote will also put a lovely equals sign, typeset by TeX, between the two figures.

[[Image:06-1350-R3-asy.png|center]]

====Outputting Equations====
We can also get useful equations out of the relation. The method <code>r3.toFormula()</code> will produce a string that is the formula for the relation.

<pre>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</pre>

This string can be written out to the standard output by <code>write(r3.toFormula())</code>. It can be written to a file by <code>file f=output("filename.txt"); write(f, r3.toFormula())</code>. The string is formatted so it can be put into TeX or a wiki page using math mode:

<math>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</math>

The method <code>r3.toLinear()</code> produces the formula in linear form:

<math>
\rho_3(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4) - b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3)
</math>

and <code>r3.toCode()</code> produces a version of the relation that can be used in Mathematica:

<pre>
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] + bp[x1, x3, x4]
- bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]
</pre>

A few relations, such as <code>r3</code>, are already defined in <code>syzygy.asy</code> but more should be added.

====Applying====
Now that we have relations, we can apply them to bigger braids. Let's start with the braid in the <math>\Phi</math> around B syzygy:

<pre>
import syzygy;
Braid b;
b.n=4;
b.add(bp,2);
b.add(bp,0);
b.add(bp,1);
b.add(bp,0);
b.add(bp,2);
b.add(phi,1);
</pre>

[[Image:06-1350-pbstart.png|center]]

After skipping the lowest knot, we can apply R3 to the next three knots:

<pre>
Braid bb=apply(r3, b, 1, 0);
</pre>

here <code>apply(r, b, k, n)</code> means we are applying the relation <code>r</code> to the braid <code>b</code> at the place in the braid found by counting <code>k</code> components up from the bottom component and <code>n</code> strands in from the leftmost strand. <code>apply</code> does not modify the original braid, but returns the result of applying the relation (stored here as <code>bb</code>):

[[Image:06-1350-pbnext.png|center]]

This went from the left hand side of the relation to the right hand side. To apply a relation in reverse, simply prefix it by a minus sign. For example <code>apply(-r3, bb, 1, 0)</code> will yield a braid equivalent to our original. When applying a relation, the script first checks that the one side of the relation matches that portion of the braid, and will give a (somewhat cryptic) error if the relation cannot be applied.

In our braids, the components are placed from bottom to top in a fixed order. Sometimes when building syzygies, it is neccessary to swap the order that these components occur. This is done by the <code>swap</code> method. For instance, starting from <code>b</code>, we can swap the two bottom crossings:

<pre>
Braid swapped=b.swap(0,1);
</pre>

[[Image:06-1350-pbswap.png|center]]

Remember that components are ordered from bottom to top, starting at 0. Again, the script checks to make sure the swap is valid (ie. changing the order of the two components, doesn't actually change the knot) and will issue an error if it isn't.

===Syzygies===
One could manually apply relations and swaps, and make a whole bunch of braids, but it would be annoying to keep track of them all. Thankfully, the <code>Syzygy</code> structure does that for us. For example, here is the complete code for the <math>\Phi</math> around B syzygy:

<pre>
import syzygy;

// Phi around B
Braid initial;
initial.n=4;
initial.add(bp,2);
initial.add(bp,0);
initial.add(bp,1);
initial.add(bp,0);
initial.add(bp,2);
initial.add(phi,1);

Syzygy pb;
pb.lsym="\Phi B";
pb.codename="PhiAroundB";
pb.initial=initial;
pb.apply(r3,1,0);
pb.apply(r4a,3,1);
pb.swap(2,3);
pb.apply(r4b,0,1);
pb.apply(-r3,1,0);
pb.apply(-r4a,0,0);
pb.swap(2,3);
pb.apply(-r4b,3,0);
pb.apply(r3,1,1);

pb.draw();
</pre>

and the result

[[Image:06-1350-PhiAroundB.png|center]]

Again, like relations, we can use <code>pb.toLinear()</code>

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

and <code>pb.toCode()</code>

<pre>
PhiAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + rho4a[x1 + x5, x2, x3, x4]
+ rho4b[x1 + x2, x3, x4, x5] - rho3[x1, x2, x3 + x4, x5] - rho4a[x1, x2, x3, x4]
- rho4b[x1, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5]
</pre>

to give the formulas for the syzygies.

The <code>Syzygy</code> structure assumes that after the last application of a relation, the braid is in the same form as the start, so it won't draw the last braid. This is annoying when building a syzygy, so it can be turned off by <code>pb.cyclic=false;</code> If you set <code>bp.showall=true;</code> the syzygy will draw all changes to the braid, including swaps. Finally, setting <code>bp.number=true;</code> will print numbers on the diagrams so you can follow them around.

===Conclusion===
Example syzygyies can be found at my [[http://www.math.utoronto.ca/andy/ lame homepage]]. Please contact me if you have any questions or suggestions. Good luck and happy syzyging!

User:Andy/06-1350-HW4

2006-12-05T22:35:42Z

Andy:

===The Generators===

Our generators are <math>T</math>, <math>R</math>, <math>\Phi</math> and <math>B^{\pm}</math>:
{| align=center cellpadding=10 style="border: solid orange 1px"
|- align=center valign=middle
|align=left|Picture
|
|
|
|[[Image:06-1350-BPlus.svg|100px]]
|
|- align=center valign=middle
|align=left|Generator
|<math>T</math>
|<math>R</math>
|<math>\Phi</math>
|<math>B^+</math>
|<math>B^-</math>
|- align=center valign=middle
|align=left|Perturbation
|<math>t</math>
|<math>r</math>
|<math>\varphi</math>
|<math>b^+</math>
|<math>b^-</math>
|}

===The Relations===

====The Reidemeister Move R2====
The following version of R2 was the easiest to use to build my [[media:06-1350-PhiAroundPhi.png|original <math>\Phi</math> around <math>\Phi</math> syzygy]]:
[[Image:06-1350-R2-weird.png|center]]
In formulas, this is
<center><math>1 = (123)^\star B^- (132)^\star B^+.</math></center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_2(x_1,x_2,x_3) = - b^-(x_1,x_2,x_3) - b^+(x_1,x_3,x_2).</math>
|}

====The Reidemeister Move R3====
The picture (with three sides of the shielding removed) is
[[Image:06-1350-R4.svg|400px|center]]
In formulas, this is
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ = (1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_3(x_1, x_2, x_3, x_4) = </math>
|<math>b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4)</math>
|-
|
|<math>- b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3).</math>
|}

====The Reidemeister Move R4====
To establish the syzygy below, I needed two versions of R4. First:
[[Image:06-1350-R4a.png|center]]
In formulas, this is
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star \Phi = (1123)^\star \Phi (1233)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_{4a}(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + \phi(x_1,x_3,x_4) - \phi(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_3+x_4).</math>
|}

Second:
[[Image:06-1350-R4b.png|center]]
In formulas, this is
<center><math>(1123)^\star B^+ (1203)^\star B^+ (1231)^\star \Phi = (1230)^\star \Phi (1223)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_{4b}(x_1,x_2,x_3,x_4) = b^+(x_1+x_2,x_3,x_4) + b^+(x_1,x_2,x_4) + \phi(x_1+x_4,x_2,x_3) - \phi(x_1,x_2,x_3) - b^+(x_1,x_2+x_3,x_4).</math>
|}

Are these independent, or can they be shown to be equivalent using other relations?

===The Syzygies===

====The "B around B" Syzygy====

The picture, with all shielding removed, is
{| align=center
|- align=center
|[[Image:06-1350-BAroundB.svg|center]]
|-
|align=right|(Drawn with [http://www.inkscape.org/ Inkscape])<br>(note that lower quality pictures are also acceptable)
|}

The functional form of this syzygy is

{| align=center
|-
|<math>BB(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1, x_2, x_3, x_5) + \rho_3(x_1 + x_5, x_2, x_3, x_4) - \rho_3(x_1 + x_2, x_3, x_4, x_5)</math>
|-
|
|<math>- \rho_3(x_1, x_2, x_4, x_5) - \rho_3(x_1 + x_4, x_2, x_3, x_5) - \rho_3(x_1, x_2, x_3, x_4)</math>
|-
|
|<math>+ \rho_3(x_1, x_3, x_4, x_5) + \rho_3(x_1 + x_3, x_2, x_4, x_5).</math>
|}

====The "<math>\Phi</math> around B" Syzygy====

The picture, with all shielding (and any other helpful notations) removed, is
{| align=center
|- align=center
|[[Image:06-1350-PhiAroundB.png|center]]
|-
|align=right|(Drawn with [http://asymptote.sf.net/ Asymptote], [[06-1350/Syzygies in Asymptote|Syzygies in Asymptote]])
|}

The functional form of this syzygy is

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

====The "<math>\Phi</math> around <math>\Phi</math>" Syzygy====

The original syzygy is available at [[:Image:06-1350-PhiAroundPhi.png]].
A cleaner, minimal picture is
{| align=center
|- align=center
|[[Image:06-1350-PhiAroundPhiClean.png|center]]
|-
|align=right|(Drawn with [http://asymptote.sf.net/ Asymptote], [[06-1350/Syzygies in Asymptote|Syzygies in Asymptote]])
|}

The functional form of this syzygy is

{|align=center
|-
|<math>\Phi\Phi(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_{4b}(x_1+x_4,x_2,x_3,x_5) + \rho_{4b}(x_1,x_2,x_3,x_4) + \rho_{4a}(x_1,x_2+x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_2,x_3,x_4+x_5) - \rho_{4a}(x_1+x_2,x_3,x_4,x_5) - \rho_{4a}(x_1,x_2,x_4,x_5).</math>
|}

===A Mathematica Verification===

The following simulated Mathematica session proves that for our single relation and single syzygy, <math>d^2=0</math>. Copy paste it into a live Mathematica session to see that it's right!

{{In|n=1|in=<nowiki>d1 = {
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] +
bp[x1, x3, x4] - bp[x1 + x2, x3, x4] - bp[x1, x2, x4] -
bp[x1 + x4, x2, x3]
};
d2 = {
BAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] +
rho3[x1 + x5, x2, x3, x4] - rho3[x1 + x2, x3, x4, x5] -
rho3[x1, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5] -
rho3[x1, x2, x3, x4] + rho3[x1, x3, x4, x5] +
rho3[x1 + x3, x2, x4, x5]
};</nowiki>}}

{{InOut|n=3|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2</nowiki>|out=<nowiki>- rho3[x1, x2, x3, x4] + rho3[x1, x2, x3, x5] - rho3[x1, x2, x4, x5]
+ rho3[x1, x3, x4, x5] - rho3[x1 + x2, x3, x4, x5]
+ rho3[x1 + x3, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5]
+ rho3[x1 + x5, x2, x3, x4]</nowiki>}}

{{InOut|n=4|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1</nowiki>|out=<nowiki>0</nowiki>}}

File:06-1350-PhiAroundPhiClean.png

2006-12-05T22:30:54Z

Andy:

User:Andy/06-1350-HW4

2006-12-05T22:30:30Z

Andy: Added cleaner Phi around Phi syzygy.

===The Generators===

Our generators are <math>T</math>, <math>R</math>, <math>\Phi</math> and <math>B^{\pm}</math>:
{| align=center cellpadding=10 style="border: solid orange 1px"
|- align=center valign=middle
|align=left|Picture
|
|
|
|[[Image:06-1350-BPlus.svg|100px]]
|
|- align=center valign=middle
|align=left|Generator
|<math>T</math>
|<math>R</math>
|<math>\Phi</math>
|<math>B^+</math>
|<math>B^-</math>
|- align=center valign=middle
|align=left|Perturbation
|<math>t</math>
|<math>r</math>
|<math>\varphi</math>
|<math>b^+</math>
|<math>b^-</math>
|}

===The Relations===

====The Reidemeister Move R2====
The following version of R2 was the easiest to use to program the <math>\Phi</math> around <math>\Phi</math> syzygy:
[[Image:06-1350-R2-weird.png|center]]
In formulas, this is
<center><math>1 = (123)^\star B^- (132)^\star B^+.</math></center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_2(x_1,x_2,x_3) = - b^-(x_1,x_2,x_3) - b^+(x_1,x_3,x_2).</math>
|}

====The Reidemeister Move R3====
The picture (with three sides of the shielding removed) is
[[Image:06-1350-R4.svg|400px|center]]
In formulas, this is
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ = (1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_3(x_1, x_2, x_3, x_4) = </math>
|<math>b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4)</math>
|-
|
|<math>- b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3).</math>
|}

====The Reidemeister Move R4====
To establish the syzygy below, I needed two versions of R4. First:
[[Image:06-1350-R4a.png|center]]
In formulas, this is
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star \Phi = (1123)^\star \Phi (1233)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_{4a}(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + \phi(x_1,x_3,x_4) - \phi(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_3+x_4).</math>
|}

Second:
[[Image:06-1350-R4b.png|center]]
In formulas, this is
<center><math>(1123)^\star B^+ (1203)^\star B^+ (1231)^\star \Phi = (1230)^\star \Phi (1223)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_{4b}(x_1,x_2,x_3,x_4) = b^+(x_1+x_2,x_3,x_4) + b^+(x_1,x_2,x_4) + \phi(x_1+x_4,x_2,x_3) - \phi(x_1,x_2,x_3) - b^+(x_1,x_2+x_3,x_4).</math>
|}

Are these independent, or can they be shown to be equivalent using other relations?

===The Syzygies===

====The "B around B" Syzygy====

The picture, with all shielding removed, is
{| align=center
|- align=center
|[[Image:06-1350-BAroundB.svg|center]]
|-
|align=right|(Drawn with [http://www.inkscape.org/ Inkscape])<br>(note that lower quality pictures are also acceptable)
|}

The functional form of this syzygy is

{| align=center
|-
|<math>BB(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1, x_2, x_3, x_5) + \rho_3(x_1 + x_5, x_2, x_3, x_4) - \rho_3(x_1 + x_2, x_3, x_4, x_5)</math>
|-
|
|<math>- \rho_3(x_1, x_2, x_4, x_5) - \rho_3(x_1 + x_4, x_2, x_3, x_5) - \rho_3(x_1, x_2, x_3, x_4)</math>
|-
|
|<math>+ \rho_3(x_1, x_3, x_4, x_5) + \rho_3(x_1 + x_3, x_2, x_4, x_5).</math>
|}

====The "<math>\Phi</math> around B" Syzygy====

The picture, with all shielding (and any other helpful notations) removed, is
{| align=center
|- align=center
|[[Image:06-1350-PhiAroundB.png|center]]
|-
|align=right|(Drawn with [http://asymptote.sf.net/ Asymptote], [[06-1350/Syzygies in Asymptote|Syzygies in Asymptote]])
|}

The functional form of this syzygy is

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

====The "<math>\Phi</math> around <math>\Phi</math>" Syzygy====

The original picture is available at [[:Image:06-1350-PhiAroundPhi.png]].
A cleaner, minimal picture is
{| align=center
|- align=center
|[[Image:06-1350-PhiAroundPhiClean.png|center]]
|-
|align=right|(Drawn with [http://asymptote.sf.net/ Asymptote], [[06-1350/Syzygies in Asymptote|Syzygies in Asymptote]])
|}

The functional form of this syzygy is

{|
|-
|<math>\Phi\Phi(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_{4b}(x_1+x_4,x_2,x_3,x_5) + \rho_{4b}(x_1,x_2,x_3,x_4) + \rho_{4a}(x_1,x_2+x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_2,x_3,x_4+x_5) - \rho_{4a}(x_1+x_2,x_3,x_4,x_5) - \rho_{4a}(x_1,x_2,x_4,x_5).</math>
|}

===A Mathematica Verification===

The following simulated Mathematica session proves that for our single relation and single syzygy, <math>d^2=0</math>. Copy paste it into a live Mathematica session to see that it's right!

{{In|n=1|in=<nowiki>d1 = {
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] +
bp[x1, x3, x4] - bp[x1 + x2, x3, x4] - bp[x1, x2, x4] -
bp[x1 + x4, x2, x3]
};
d2 = {
BAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] +
rho3[x1 + x5, x2, x3, x4] - rho3[x1 + x2, x3, x4, x5] -
rho3[x1, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5] -
rho3[x1, x2, x3, x4] + rho3[x1, x3, x4, x5] +
rho3[x1 + x3, x2, x4, x5]
};</nowiki>}}

{{InOut|n=3|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2</nowiki>|out=<nowiki>- rho3[x1, x2, x3, x4] + rho3[x1, x2, x3, x5] - rho3[x1, x2, x4, x5]
+ rho3[x1, x3, x4, x5] - rho3[x1 + x2, x3, x4, x5]
+ rho3[x1 + x3, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5]
+ rho3[x1 + x5, x2, x3, x4]</nowiki>}}

{{InOut|n=4|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1</nowiki>|out=<nowiki>0</nowiki>}}

User:Andy/06-1350-HW4

2006-12-05T21:53:41Z

Andy: /* The "<math>\Phi</math> around <math>\Phi</math>" Syzygy */

===The Generators===

Our generators are <math>T</math>, <math>R</math>, <math>\Phi</math> and <math>B^{\pm}</math>:
{| align=center cellpadding=10 style="border: solid orange 1px"
|- align=center valign=middle
|align=left|Picture
|
|
|
|[[Image:06-1350-BPlus.svg|100px]]
|
|- align=center valign=middle
|align=left|Generator
|<math>T</math>
|<math>R</math>
|<math>\Phi</math>
|<math>B^+</math>
|<math>B^-</math>
|- align=center valign=middle
|align=left|Perturbation
|<math>t</math>
|<math>r</math>
|<math>\varphi</math>
|<math>b^+</math>
|<math>b^-</math>
|}

===The Relations===

====The Reidemeister Move R2====
The following version of R2 was the easiest to use to program the <math>\Phi</math> around <math>\Phi</math> syzygy:
[[Image:06-1350-R2-weird.png|center]]
In formulas, this is
<center><math>1 = (123)^\star B^- (132)^\star B^+.</math></center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_2(x_1,x_2,x_3) = - b^-(x_1,x_2,x_3) - b^+(x_1,x_3,x_2).</math>
|}

====The Reidemeister Move R3====
The picture (with three sides of the shielding removed) is
[[Image:06-1350-R4.svg|400px|center]]
In formulas, this is
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ = (1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_3(x_1, x_2, x_3, x_4) = </math>
|<math>b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4)</math>
|-
|
|<math>- b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3).</math>
|}

====The Reidemeister Move R4====
To establish the syzygy below, I needed two versions of R4. First:
[[Image:06-1350-R4a.png|center]]
In formulas, this is
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star \Phi = (1123)^\star \Phi (1233)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_{4a}(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + \phi(x_1,x_3,x_4) - \phi(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_3+x_4).</math>
|}

Second:
[[Image:06-1350-R4b.png|center]]
In formulas, this is
<center><math>(1123)^\star B^+ (1203)^\star B^+ (1231)^\star \Phi = (1230)^\star \Phi (1223)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_{4b}(x_1,x_2,x_3,x_4) = b^+(x_1+x_2,x_3,x_4) + b^+(x_1,x_2,x_4) + \phi(x_1+x_4,x_2,x_3) - \phi(x_1,x_2,x_3) - b^+(x_1,x_2+x_3,x_4).</math>
|}

Are these independent, or can they be shown to be equivalent using other relations?

===The Syzygies===

====The "B around B" Syzygy====

The picture, with all shielding removed, is
{| align=center
|- align=center
|[[Image:06-1350-BAroundB.svg|center]]
|-
|align=right|(Drawn with [http://www.inkscape.org/ Inkscape])<br>(note that lower quality pictures are also acceptable)
|}

The functional form of this syzygy is

{| align=center
|-
|<math>BB(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1, x_2, x_3, x_5) + \rho_3(x_1 + x_5, x_2, x_3, x_4) - \rho_3(x_1 + x_2, x_3, x_4, x_5)</math>
|-
|
|<math>- \rho_3(x_1, x_2, x_4, x_5) - \rho_3(x_1 + x_4, x_2, x_3, x_5) - \rho_3(x_1, x_2, x_3, x_4)</math>
|-
|
|<math>+ \rho_3(x_1, x_3, x_4, x_5) + \rho_3(x_1 + x_3, x_2, x_4, x_5).</math>
|}

====The "<math>\Phi</math> around B" Syzygy====

The picture, with all shielding (and any other helpful notations) removed, is
{| align=center
|- align=center
|[[Image:06-1350-PhiAroundB.png|center]]
|-
|align=right|(Drawn with [http://asymptote.sf.net/ Asymptote], [[06-1350/Syzygies in Asymptote|Syzygies in Asymptote]])
|}

The functional form of this syzygy is

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

====The "<math>\Phi</math> around <math>\Phi</math>" Syzygy====

The picture is
{| align=center
|- align=center
|[[Image:06-1350-PhiAroundPhi.png|center]]
|-
|align=right|(Drawn with [http://asymptote.sf.net/ Asymptote], [[06-1350/Syzygies in Asymptote|Syzygies in Asymptote]])
|}

The functional form of this syzygy is

{| align=center
|-
|<math>\Phi\Phi(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_2(x_1+x_2,x_3,x_4) + \rho_2(x_1+x_2+x_4,x_3,x_5) + \rho_{4b}(x_1+x_2,x_4,x_5,x_3)</math>
|-
|
|<math>+ \rho_2(x_1,x_2,x_4) + \rho_2(x_1+x_4,x_2,x_5) + \rho_{4b}(x_1,x_4,x_5,x_2)</math>
|-
|
|<math>+ \rho_{4a}(x_1,x_4+x_5,x_2,x_3) - \rho_{4b}(x_1,x_4,x_5,x_2+x_3) - \rho_{4a}(x_1+x_4,x_5,x_2,x_3)</math>
|-
|
|<math>- \rho_2(x_1+x_4,x_2,x_5) - \rho_2(x_1+x_2+x_4,x_3,x_5) - \rho_{4a}(x_1,x_4,x_2,x_3)</math>
|-
|
|<math>- \rho_2(x_1,x_2,x_4) - \rho_2(x_1+x_2,x_3,x_4)</math>
|}

Note that the first and last terms cancel, as the two steps at the top of the diagram are opposites.

===A Mathematica Verification===

The following simulated Mathematica session proves that for our single relation and single syzygy, <math>d^2=0</math>. Copy paste it into a live Mathematica session to see that it's right!

{{In|n=1|in=<nowiki>d1 = {
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] +
bp[x1, x3, x4] - bp[x1 + x2, x3, x4] - bp[x1, x2, x4] -
bp[x1 + x4, x2, x3]
};
d2 = {
BAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] +
rho3[x1 + x5, x2, x3, x4] - rho3[x1 + x2, x3, x4, x5] -
rho3[x1, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5] -
rho3[x1, x2, x3, x4] + rho3[x1, x3, x4, x5] +
rho3[x1 + x3, x2, x4, x5]
};</nowiki>}}

{{InOut|n=3|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2</nowiki>|out=<nowiki>- rho3[x1, x2, x3, x4] + rho3[x1, x2, x3, x5] - rho3[x1, x2, x4, x5]
+ rho3[x1, x3, x4, x5] - rho3[x1 + x2, x3, x4, x5]
+ rho3[x1 + x3, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5]
+ rho3[x1 + x5, x2, x3, x4]</nowiki>}}

{{InOut|n=4|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1</nowiki>|out=<nowiki>0</nowiki>}}

File:06-1350-R2-weird.png

2006-12-05T19:30:59Z

Andy:

User:Andy/06-1350-HW4

2006-12-05T19:30:23Z

Andy: Added weird R2.

===The Generators===

Our generators are <math>T</math>, <math>R</math>, <math>\Phi</math> and <math>B^{\pm}</math>:
{| align=center cellpadding=10 style="border: solid orange 1px"
|- align=center valign=middle
|align=left|Picture
|
|
|
|[[Image:06-1350-BPlus.svg|100px]]
|
|- align=center valign=middle
|align=left|Generator
|<math>T</math>
|<math>R</math>
|<math>\Phi</math>
|<math>B^+</math>
|<math>B^-</math>
|- align=center valign=middle
|align=left|Perturbation
|<math>t</math>
|<math>r</math>
|<math>\varphi</math>
|<math>b^+</math>
|<math>b^-</math>
|}

===The Relations===

====The Reidemeister Move R2====
The following version of R2 was the easiest to use to program the <math>\Phi</math> around <math>\Phi</math> syzygy:
[[Image:06-1350-R2-weird.png|center]]
In formulas, this is
<center><math>1 = (123)^\star B^- (132)^\star B^+.</math></center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_2(x_1,x_2,x_3) = - b^-(x_1,x_2,x_3) - b^+(x_1,x_3,x_2).</math>
|}

====The Reidemeister Move R3====
The picture (with three sides of the shielding removed) is
[[Image:06-1350-R4.svg|400px|center]]
In formulas, this is
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ = (1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_3(x_1, x_2, x_3, x_4) = </math>
|<math>b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4)</math>
|-
|
|<math>- b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3).</math>
|}

====The Reidemeister Move R4====
To establish the syzygy below, I needed two versions of R4. First:
[[Image:06-1350-R4a.png|center]]
In formulas, this is
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star \Phi = (1123)^\star \Phi (1233)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_{4a}(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + \phi(x_1,x_3,x_4) - \phi(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_3+x_4).</math>
|}

Second:
[[Image:06-1350-R4b.png|center]]
In formulas, this is
<center><math>(1123)^\star B^+ (1203)^\star B^+ (1231)^\star \Phi = (1230)^\star \Phi (1223)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_{4b}(x_1,x_2,x_3,x_4) = b^+(x_1+x_2,x_3,x_4) + b^+(x_1,x_2,x_4) + \phi(x_1+x_4,x_2,x_3) - \phi(x_1,x_2,x_3) - b^+(x_1,x_2+x_3,x_4).</math>
|}

Are these independent, or can they be shown to be equivalent using other relations?

===The Syzygies===

====The "B around B" Syzygy====

The picture, with all shielding removed, is
{| align=center
|- align=center
|[[Image:06-1350-BAroundB.svg|center]]
|-
|align=right|(Drawn with [http://www.inkscape.org/ Inkscape])<br>(note that lower quality pictures are also acceptable)
|}

The functional form of this syzygy is

{| align=center
|-
|<math>BB(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1, x_2, x_3, x_5) + \rho_3(x_1 + x_5, x_2, x_3, x_4) - \rho_3(x_1 + x_2, x_3, x_4, x_5)</math>
|-
|
|<math>- \rho_3(x_1, x_2, x_4, x_5) - \rho_3(x_1 + x_4, x_2, x_3, x_5) - \rho_3(x_1, x_2, x_3, x_4)</math>
|-
|
|<math>+ \rho_3(x_1, x_3, x_4, x_5) + \rho_3(x_1 + x_3, x_2, x_4, x_5).</math>
|}

====The "<math>\Phi</math> around B" Syzygy====

The picture, with all shielding (and any other helpful notations) removed, is
{| align=center
|- align=center
|[[Image:06-1350-PhiAroundB.png|center]]
|-
|align=right|(Drawn with [http://asymptote.sf.net/ Asymptote], [[06-1350/Syzygies in Asymptote|Syzygies in Asymptote]])
|}

The functional form of this syzygy is

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

====The "<math>\Phi</math> around <math>\Phi</math>" Syzygy====

The picture is
{| align=center
|- align=center
|[[Image:06-1350-PhiAroundPhi.png|center]]
|-
|align=right|(Drawn with [http://asymptote.sf.net/ Asymptote], [[06-1350/Syzygies in Asymptote|Syzygies in Asymptote]])
|}

The functional form of this syzygy is

{| align=center
|-
|<math>\Phi\Phi(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_2(x_1+x_2,x_3,x_4) + \rho_2(x_1+x_2+x_4,x_3,x_5) + \rho_{4b}(x_1+x_2,x_4,x_5,x_3)</math>
|-
|
|<math>+ \rho_2(x_1,x_2,x_4) + \rho_2(x_1+x_4,x_2,x_5) + \rho_{4b}(x_1,x_4,x_5,x_2)</math>
|-
|
|<math>+ \rho_{4a}(x_1,x_4+x_5,x_2,x_3) - \rho_{4b}(x_1,x_4,x_5,x_2+x_3) - \rho_{4a}(x_1+x_4,x_5,x_2,x_3)</math>
|-
|
|<math>- \rho_2(x_1+x_4,x_2,x_5) - \rho_2(x_1+x_2+x_4,x_3,x_5) - \rho_{4a}(x_1,x_4,x_2,x_3)</math>
|-
|
|<math>- \rho_2(x_1,x_2,x_4) - \rho_2(x_1+x_2,x_3,x_4)</math>
|}

Note that the first and last terms cancel, as the top steps at the top of the diagram are opposites.

===A Mathematica Verification===

The following simulated Mathematica session proves that for our single relation and single syzygy, <math>d^2=0</math>. Copy paste it into a live Mathematica session to see that it's right!

{{In|n=1|in=<nowiki>d1 = {
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] +
bp[x1, x3, x4] - bp[x1 + x2, x3, x4] - bp[x1, x2, x4] -
bp[x1 + x4, x2, x3]
};
d2 = {
BAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] +
rho3[x1 + x5, x2, x3, x4] - rho3[x1 + x2, x3, x4, x5] -
rho3[x1, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5] -
rho3[x1, x2, x3, x4] + rho3[x1, x3, x4, x5] +
rho3[x1 + x3, x2, x4, x5]
};</nowiki>}}

{{InOut|n=3|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2</nowiki>|out=<nowiki>- rho3[x1, x2, x3, x4] + rho3[x1, x2, x3, x5] - rho3[x1, x2, x4, x5]
+ rho3[x1, x3, x4, x5] - rho3[x1 + x2, x3, x4, x5]
+ rho3[x1 + x3, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5]
+ rho3[x1 + x5, x2, x3, x4]</nowiki>}}

{{InOut|n=4|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1</nowiki>|out=<nowiki>0</nowiki>}}

File:06-1350-PhiAroundPhi.png

2006-12-05T19:15:08Z

Andy:

User:Andy/06-1350-HW4

2006-12-05T19:14:36Z

Andy: Added Phi around Phi syzygy.

===The Generators===

Our generators are <math>T</math>, <math>R</math>, <math>\Phi</math> and <math>B^{\pm}</math>:
{| align=center cellpadding=10 style="border: solid orange 1px"
|- align=center valign=middle
|align=left|Picture
|
|
|
|[[Image:06-1350-BPlus.svg|100px]]
|
|- align=center valign=middle
|align=left|Generator
|<math>T</math>
|<math>R</math>
|<math>\Phi</math>
|<math>B^+</math>
|<math>B^-</math>
|- align=center valign=middle
|align=left|Perturbation
|<math>t</math>
|<math>r</math>
|<math>\varphi</math>
|<math>b^+</math>
|<math>b^-</math>
|}

===The Relations===

====The Reidemeister Move R3====
The picture (with three sides of the shielding removed) is
[[Image:06-1350-R4.svg|400px|center]]
In formulas, this is
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ = (1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_3(x_1, x_2, x_3, x_4) = </math>
|<math>b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4)</math>
|-
|
|<math>- b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3).</math>
|}

====The Reidemeister Move R4====
To establish the syzygy below, I needed two versions of R4. First:
[[Image:06-1350-R4a.png|center]]
In formulas, this is
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star \Phi = (1123)^\star \Phi (1233)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_{4a}(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + \phi(x_1,x_3,x_4) - \phi(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_3+x_4).</math>
|}

Second:
[[Image:06-1350-R4b.png|center]]
In formulas, this is
<center><math>(1123)^\star B^+ (1203)^\star B^+ (1231)^\star \Phi = (1230)^\star \Phi (1223)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_{4b}(x_1,x_2,x_3,x_4) = b^+(x_1+x_2,x_3,x_4) + b^+(x_1,x_2,x_4) + \phi(x_1+x_4,x_2,x_3) - \phi(x_1,x_2,x_3) - b^+(x_1,x_2+x_3,x_4).</math>
|}

Are these independent, or can they be shown to be equivalent using other relations?

===The Syzygies===

====The "B around B" Syzygy====

The picture, with all shielding removed, is
{| align=center
|- align=center
|[[Image:06-1350-BAroundB.svg|center]]
|-
|align=right|(Drawn with [http://www.inkscape.org/ Inkscape])<br>(note that lower quality pictures are also acceptable)
|}

The functional form of this syzygy is

{| align=center
|-
|<math>BB(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1, x_2, x_3, x_5) + \rho_3(x_1 + x_5, x_2, x_3, x_4) - \rho_3(x_1 + x_2, x_3, x_4, x_5)</math>
|-
|
|<math>- \rho_3(x_1, x_2, x_4, x_5) - \rho_3(x_1 + x_4, x_2, x_3, x_5) - \rho_3(x_1, x_2, x_3, x_4)</math>
|-
|
|<math>+ \rho_3(x_1, x_3, x_4, x_5) + \rho_3(x_1 + x_3, x_2, x_4, x_5).</math>
|}

====The "<math>\Phi</math> around B" Syzygy====

The picture, with all shielding (and any other helpful notations) removed, is
{| align=center
|- align=center
|[[Image:06-1350-PhiAroundB.png|center]]
|-
|align=right|(Drawn with [http://asymptote.sf.net/ Asymptote], [[06-1350/Syzygies in Asymptote|Syzygies in Asymptote]])
|}

The functional form of this syzygy is

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

====The "<math>\Phi</math> around <math>\Phi</math>" Syzygy====

The picture is
{| align=center
|- align=center
|[[Image:06-1350-PhiAroundPhi.png|center]]
|-
|align=right|(Drawn with [http://asymptote.sf.net/ Asymptote], [[06-1350/Syzygies in Asymptote|Syzygies in Asymptote]])
|}

The functional form of this syzygy is

{| align=center
|-
|<math>\Phi\Phi(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_2(x_1+x_2,x_3,x_4) + \rho_2(x_1+x_2+x_4,x_3,x_5) + \rho_{4b}(x_1+x_2,x_4,x_5,x_3)</math>
|-
|
|<math>+ \rho_2(x_1,x_2,x_4) + \rho_2(x_1+x_4,x_2,x_5) + \rho_{4b}(x_1,x_4,x_5,x_2)</math>
|-
|
|<math>+ \rho_{4a}(x_1,x_4+x_5,x_2,x_3) - \rho_{4b}(x_1,x_4,x_5,x_2+x_3) - \rho_{4a}(x_1+x_4,x_5,x_2,x_3)</math>
|-
|
|<math>- \rho_2(x_1+x_4,x_2,x_5) - \rho_2(x_1+x_2+x_4,x_3,x_5) - \rho_{4a}(x_1,x_4,x_2,x_3)</math>
|-
|
|<math>- \rho_2(x_1,x_2,x_4) - \rho_2(x_1+x_2,x_3,x_4)</math>
|}

Note that the first and last terms cancel, as the top steps at the top of the diagram are opposites.

===A Mathematica Verification===

The following simulated Mathematica session proves that for our single relation and single syzygy, <math>d^2=0</math>. Copy paste it into a live Mathematica session to see that it's right!

{{In|n=1|in=<nowiki>d1 = {
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] +
bp[x1, x3, x4] - bp[x1 + x2, x3, x4] - bp[x1, x2, x4] -
bp[x1 + x4, x2, x3]
};
d2 = {
BAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] +
rho3[x1 + x5, x2, x3, x4] - rho3[x1 + x2, x3, x4, x5] -
rho3[x1, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5] -
rho3[x1, x2, x3, x4] + rho3[x1, x3, x4, x5] +
rho3[x1 + x3, x2, x4, x5]
};</nowiki>}}

{{InOut|n=3|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2</nowiki>|out=<nowiki>- rho3[x1, x2, x3, x4] + rho3[x1, x2, x3, x5] - rho3[x1, x2, x4, x5]
+ rho3[x1, x3, x4, x5] - rho3[x1 + x2, x3, x4, x5]
+ rho3[x1 + x3, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5]
+ rho3[x1 + x5, x2, x3, x4]</nowiki>}}

{{InOut|n=4|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1</nowiki>|out=<nowiki>0</nowiki>}}

06-1350/Syzygies in Asymptote in Brief

2006-12-05T00:29:34Z

Andy:

===Installation===
See [[06-1350/Syzygies in Asymptote]] for more detailed information.

First install [http://asymptote.sourceforge.net Asymptote]. Once installed, download [http://www.math.utoronto.ca/~andy/syzygy.asy syzygy.asy].

===Braids===
{|align=center cellpadding="5"
|-
|<pre>
import syzygy; // Accesses the syzygy module.
Braid b; // Start a new braid.
b.n=3; // The braid has three strands.
// The strands are numbered left to right starting at 0.
b.add(bp,0); // Add a overcrossing component starting at strand 0,
// the leftmost strand.
b.add(bm,1); // Add an undercrossing starting at strand 1.
b.add(phi,0); // Add a trivalent vertex that merges strands 0 and 1.
// Strand 2 is now renumbered as strand 1.
b.draw(); // Draw the resulting braid.
</pre>
|[[Image:06-1350-mybraid.png]]
|}

===Relations===
{|align=center cellpadding="5"
|-
|<pre>
import syzygy; // Access the syzygy module.
Braid l; // Define the left hand side of the relation.
l.n=3; l.add(bp,0); l.add(bp,1); l.add(bp,0);
Braid r; // Define the right hand side of the relation.
r.n=3; r.add(bp,1); r.add(bp,0); r.add(bp,1);

Relation r3; // Define a relation.
r3.lsym="\rho_3"; // Give the relation a formula name.
r3.codename="rho3"; // Give the relation a name to be used by Mathematica.
r3.lhs=l; r3.rhs=r;
r3.draw();
</pre>
|[[Image:06-1350-R3-asy.png]]
|}

<code>r3.toFormula()</code> produces the formula:
<center><math>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</math></center>

<code>r3.toLinear()</code> produces the formula in linear form:

{|align=center
|-
|<math>\rho_3(x_1,x_2,x_3,x_4) = </math>
|<math>b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4)</math>
|-
|
|<math>- b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3)</math>
|}

and <code>r3.toCode()</code> produces a version usable in Mathematica:

<pre>
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] + bp[x1, x3, x4]
- bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]
</pre>

===Syzygies===

{|align=center cellpadding="5"
|-
|<pre>
import syzygy;

// Phi around B
Braid initial;
initial.n=4;
initial.add(bp,2);
initial.add(bp,0);
initial.add(bp,1);
initial.add(bp,0);
initial.add(bp,2);
initial.add(phi,1);

Syzygy pb;
pb.lsym="\Phi B";
pb.codename="PhiAroundB";
pb.initial=initial;
pb.apply(r3,1,0);
pb.apply(r4a,3,1);
pb.swap(2,3);
pb.apply(r4b,0,1);
pb.apply(-r3,1,0);
pb.apply(-r4a,0,0);
pb.swap(2,3);
pb.apply(-r4b,3,0);
pb.apply(r3,1,1);

pb.draw();
</pre>
|[[Image:06-1350-bpsmall.png]]
|}

Again, like relations, we can use <code>pb.toLinear()</code>

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

and <code>pb.toCode()</code>

<pre>
PhiAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + rho4a[x1 + x5, x2, x3, x4]
+ rho4b[x1 + x2, x3, x4, x5] - rho3[x1, x2, x3 + x4, x5] - rho4a[x1, x2, x3, x4]
- rho4b[x1, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5]
</pre>

to produce formulas.

06-1350/Syzygies in Asymptote in Brief

2006-12-05T00:25:11Z

Andy:

===Installation===
See [[06-1350/Syzygies in Asymptote]] for more detailed information.

First install [http://asymptote.sourceforge.net Asymptote]. Once installed, download [http://www.math.utoronto.ca/~andy/syzygy.asy syzygy.asy].

===Braids===
{|align=center cellpadding="5"
|-
|<pre>
import syzygy; // Accesses the syzygy module.
Braid b; // Start a new braid.
b.n=3; // The braid has three strands.
// The strands are numbered left to right starting at 0.
b.add(bp,0); // Add a overcrossing component starting at strand 0,
// the leftmost strand.
b.add(bm,1); // Add an undercrossing starting at strand 1.
b.add(phi,0); // Add a trivalent vertex that merges strands 0 and 1.
// Strand 2 is now renumbered as strand 1.
b.draw(); // Draw the resulting braid.
</pre>
|[[Image:06-1350-mybraid.png]]
|}

===Relations===
{|align=center cellpadding="5"
|-
|<pre>
import syzygy; // Access the syzygy module.
Braid l; // Define the left hand side of the relation.
l.n=3; l.add(bp,0); l.add(bp,1); l.add(bp,0);
Braid r; // Define the right hand side of the relation.
r.n=3; r.add(bp,1); r.add(bp,0); r.add(bp,1);

Relation r3; // Define a relation.
r3.lsym="\rho_3"; // Give the relation a formula name.
r3.codename="rho3"; // Give the relation a name to be used by Mathematica.
r3.lhs=l; r3.rhs=r;
r3.draw();
</pre>
|[[Image:06-1350-R3-asy.png]]
|}

<code>r3.toFormula()</code> produces the formula:

<math>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</math>

<code>r3.toLinear()</code> produces the formula in linear form:

<math>
\rho_3(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4) - b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3)
</math>

and <code>r3.toCode()</code> produces a version usable in Mathematica:

<pre>
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] + bp[x1, x3, x4]
- bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]
</pre>

===Syzygies===

{|align=center cellpadding="5"
|-
|<pre>
import syzygy;

// Phi around B
Braid initial;
initial.n=4;
initial.add(bp,2);
initial.add(bp,0);
initial.add(bp,1);
initial.add(bp,0);
initial.add(bp,2);
initial.add(phi,1);

Syzygy pb;
pb.lsym="\Phi B";
pb.codename="PhiAroundB";
pb.initial=initial;
pb.apply(r3,1,0);
pb.apply(r4a,3,1);
pb.swap(2,3);
pb.apply(r4b,0,1);
pb.apply(-r3,1,0);
pb.apply(-r4a,0,0);
pb.swap(2,3);
pb.apply(-r4b,3,0);
pb.apply(r3,1,1);

pb.draw();
</pre>
|[[Image:06-1350-bpsmall.png]]
|}

Again, like relations, we can use <code>pb.toLinear()</code>

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

and <code>pb.toCode()</code>

<pre>
PhiAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + rho4a[x1 + x5, x2, x3, x4]
+ rho4b[x1 + x2, x3, x4, x5] - rho3[x1, x2, x3 + x4, x5] - rho4a[x1, x2, x3, x4]
- rho4b[x1, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5]
</pre>

to produce formulas.

06-1350/Syzygies in Asymptote in Brief

2006-12-05T00:23:33Z

Andy: Formatting.

===Installation===
See [[06-1350/Syzygies in Asymptote]] for more detailed information.

First install [http://asymptote.sourceforge.net Asymptote]. Once installed, download [http://www.math.utoronto.ca/~andy/syzygy.asy syzygy.asy].

===Braids===
{|align=center cellpadding="20"
|-
|<pre>
import syzygy; // Accesses the syzygy module.
Braid b; // Start a new braid.
b.n=3; // The braid has three strands.
// The strands are numbered left to right starting at 0.
b.add(bp,0); // Add a overcrossing component starting at strand 0,
// the leftmost strand.
b.add(bm,1); // Add an undercrossing starting at strand 1.
b.add(phi,0); // Add a trivalent vertex that merges strands 0 and 1.
// Strand 2 is now renumbered as strand 1.
b.draw(); // Draw the resulting braid.
</pre>
|[[Image:06-1350-mybraid.png]]
|}

===Relations===
{|align=center cellpadding="20"
|-
|<pre>
import syzygy; // Access the syzygy module.
Braid l; // Define the left hand side of the relation.
l.n=3; l.add(bp,0); l.add(bp,1); l.add(bp,0);
Braid r; // Define the right hand side of the relation.
r.n=3; r.add(bp,1); r.add(bp,0); r.add(bp,1);

Relation r3; // Define a relation.
r3.lsym="\rho_3"; // Give the relation a formula name.
r3.codename="rho3"; // Give the relation a name to be used by Mathematica.
r3.lhs=l; r3.rhs=r;
r3.draw();
</pre>
|[[Image:06-1350-R3-asy.png]]
|}

<code>r3.toFormula()</code> produces the formula:

<math>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</math>

<code>r3.toLinear()</code> produces the formula in linear form:

<math>
\rho_3(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4) - b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3)
</math>

and <code>r3.toCode()</code> produces a version usable in Mathematica:

<pre>
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] + bp[x1, x3, x4]
- bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]
</pre>

===Syzygies===

{|align=center cellpadding="20"
|-
|<pre>
import syzygy;

// Phi around B
Braid initial;
initial.n=4;
initial.add(bp,2);
initial.add(bp,0);
initial.add(bp,1);
initial.add(bp,0);
initial.add(bp,2);
initial.add(phi,1);

Syzygy pb;
pb.lsym="\Phi B";
pb.codename="PhiAroundB";
pb.initial=initial;
pb.apply(r3,1,0);
pb.apply(r4a,3,1);
pb.swap(2,3);
pb.apply(r4b,0,1);
pb.apply(-r3,1,0);
pb.apply(-r4a,0,0);
pb.swap(2,3);
pb.apply(-r4b,3,0);
pb.apply(r3,1,1);

pb.draw();
</pre>
|[[Image:06-1350-bpsmall.png]]
|}

Again, like relations, we can use <code>pb.toLinear()</code>

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

and <code>pb.toCode()</code>

<pre>
PhiAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + rho4a[x1 + x5, x2, x3, x4]
+ rho4b[x1 + x2, x3, x4, x5] - rho3[x1, x2, x3 + x4, x5] - rho4a[x1, x2, x3, x4]
- rho4b[x1, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5]
</pre>

to produce formulas.

File:06-1350-bpsmall.png

2006-12-04T23:48:32Z

Andy:

06-1350/Syzygies in Asymptote in Brief

2006-12-04T23:48:05Z

Andy:

===Installation===
See [[06-1350/Syzygies in Asymptote]] for more detailed information.

First install [http://asymptote.sourceforge.net Asymptote]. Once installed, download [http://www.math.utoronto.ca/~andy/syzygy.asy syzygy.asy].

===Braids===
[[Image:06-1350-mybraid.png|frame]]
<pre>
import syzygy; // Accesses the syzygy module.
Braid b; // Start a new braid.
b.n=3; // The braid has three strands.
// The strands are numbered left to right starting at 0.
b.add(bp,0); // Add a overcrossing component starting at strand 0,
// the leftmost strand.
b.add(bm,1); // Add an undercrossing starting at strand 1.
b.add(phi,0); // Add a trivalent vertex that merges strands 0 and 1.
// Strand 2 is now renumbered as strand 1.
b.draw(); // Draw the resulting braid.
</pre>

===Relations===
[[Image:06-1350-R3-asy.png|frame]]
<pre>
import syzygy; // Access the syzygy module.
Braid l; // Define the left hand side of the relation.
l.n=3; l.add(bp,0); l.add(bp,1); l.add(bp,0);
Braid r; // Define the right hand side of the relation.
r.n=3; r.add(bp,1); r.add(bp,0); r.add(bp,1);

Relation r3; // Define a relation.
r3.lsym="\rho_3"; // Give the relation a name for when it is written in functional form.
r3.codename="rho3"; // Give the relation a name to be used by Mathematica.
r3.lhs=l; r3.rhs=r;
r3.draw();
</pre>

<code>r3.toFormula()</code> produces the formula:

<math>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</math>

<code>r3.toLinear()</code> produces the formula in linear form:

<math>
\rho_3(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4) - b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3)
</math>

and <code>r3.toCode()</code> produces a version usable in Mathematica:

<pre>
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] + bp[x1, x3, x4]
- bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]
</pre>

===Syzygies===

[[Image:06-1350-bpsmall.png|frame]]
<pre>
import syzygy;

// Phi around B
Braid initial;
initial.n=4;
initial.add(bp,2);
initial.add(bp,0);
initial.add(bp,1);
initial.add(bp,0);
initial.add(bp,2);
initial.add(phi,1);

Syzygy pb;
pb.lsym="\Phi B";
pb.codename="PhiAroundB";
pb.initial=initial;
pb.apply(r3,1,0);
pb.apply(r4a,3,1);
pb.swap(2,3);
pb.apply(r4b,0,1);
pb.apply(-r3,1,0);
pb.apply(-r4a,0,0);
pb.swap(2,3);
pb.apply(-r4b,3,0);
pb.apply(r3,1,1);

pb.draw();
</pre>

Again, like relations, we can use <code>pb.toLinear()</code>

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

and <code>pb.toCode()</code>

<pre>
PhiAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + rho4a[x1 + x5, x2, x3, x4]
+ rho4b[x1 + x2, x3, x4, x5] - rho3[x1, x2, x3 + x4, x5] - rho4a[x1, x2, x3, x4]
- rho4b[x1, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5]
</pre>

to produce formulas.

06-1350/Syzygies in Asymptote in Brief

2006-12-04T23:37:01Z

Andy: /* Installation */

===Installation===
See [[06-1350/Syzygies in Asymptote]] for more detailed information.

First install [http://asymptote.sourceforge.net Asymptote]. Once installed, download [http://www.math.utoronto.ca/~andy/syzygy.asy syzygy.asy].

===Braids===
<pre>
import syzygy; // Accesses the syzygy module.
Braid b; // Start a new braid.
b.n=3; // The braid has three strands.
// The strands are numbered left to right starting at 0.
b.add(bp,0); // Add a overcrossing component starting at strand 0,
// the leftmost strand.
b.add(bm,1); // Add an undercrossing starting at strand 1.
b.add(phi,0); // Add a trivalent vertex that merges strands 0 and 1.
// Strand 2 is now renumbered as strand 1.
b.draw(); // Draw the resulting braid.
</pre>

[[Image:06-1350-mybraid.png|center]]

===Relations===
<pre>
import syzygy; // Access the syzygy module.
Braid l; // Define the left hand side of the relation.
l.n=3; l.add(bp,0); l.add(bp,1); l.add(bp,0);
Braid r; // Define the right hand side of the relation.
r.n=3; r.add(bp,1); r.add(bp,0); r.add(bp,1);

Relation r3; // Define a relation.
r3.lsym="\rho_3"; // Give the relation a name for when it is written in functional form.
r3.codename="rho3"; // Give the relation a name to be used by Mathematica.
r3.lhs=l; r3.rhs=r;
r3.draw();
</pre>

[[Image:06-1350-R3-asy.png|center]]

<code>r3.toFormula()</code> produces the formula:

<math>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</math>

<code>r3.toLinear()</code> produces the formula in linear form:

<math>
\rho_3(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4) - b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3)
</math>

and <code>r3.toCode()</code> produces a version usable in Mathematica:

<pre>
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] + bp[x1, x3, x4]
- bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]
</pre>

===Syzygies===
<pre>
import syzygy;

// Phi around B
Braid initial;
initial.n=4;
initial.add(bp,2);
initial.add(bp,0);
initial.add(bp,1);
initial.add(bp,0);
initial.add(bp,2);
initial.add(phi,1);

Syzygy pb;
pb.lsym="\Phi B";
pb.codename="PhiAroundB";
pb.initial=initial;
pb.apply(r3,1,0);
pb.apply(r4a,3,1);
pb.swap(2,3);
pb.apply(r4b,0,1);
pb.apply(-r3,1,0);
pb.apply(-r4a,0,0);
pb.swap(2,3);
pb.apply(-r4b,3,0);
pb.apply(r3,1,1);

pb.draw();
</pre>

[[Image:06-1350-PhiAroundB.png|center]]

Again, like relations, we can use <code>pb.toLinear()</code>

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

and <code>pb.toCode()</code>

<pre>
PhiAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + rho4a[x1 + x5, x2, x3, x4]
+ rho4b[x1 + x2, x3, x4, x5] - rho3[x1, x2, x3 + x4, x5] - rho4a[x1, x2, x3, x4]
- rho4b[x1, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5]
</pre>

to produce formulas.

06-1350/Syzygies in Asymptote in Brief

2006-12-04T23:36:32Z

Andy: For Handout.

===Installation===
See [[Syzygies in Asymptote]] for more detailed information.

First install [http://asymptote.sourceforge.net Asymptote]. Once installed, download [http://www.math.utoronto.ca/~andy/syzygy.asy syzygy.asy].

===Braids===
<pre>
import syzygy; // Accesses the syzygy module.
Braid b; // Start a new braid.
b.n=3; // The braid has three strands.
// The strands are numbered left to right starting at 0.
b.add(bp,0); // Add a overcrossing component starting at strand 0,
// the leftmost strand.
b.add(bm,1); // Add an undercrossing starting at strand 1.
b.add(phi,0); // Add a trivalent vertex that merges strands 0 and 1.
// Strand 2 is now renumbered as strand 1.
b.draw(); // Draw the resulting braid.
</pre>

[[Image:06-1350-mybraid.png|center]]

===Relations===
<pre>
import syzygy; // Access the syzygy module.
Braid l; // Define the left hand side of the relation.
l.n=3; l.add(bp,0); l.add(bp,1); l.add(bp,0);
Braid r; // Define the right hand side of the relation.
r.n=3; r.add(bp,1); r.add(bp,0); r.add(bp,1);

Relation r3; // Define a relation.
r3.lsym="\rho_3"; // Give the relation a name for when it is written in functional form.
r3.codename="rho3"; // Give the relation a name to be used by Mathematica.
r3.lhs=l; r3.rhs=r;
r3.draw();
</pre>

[[Image:06-1350-R3-asy.png|center]]

<code>r3.toFormula()</code> produces the formula:

<math>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</math>

<code>r3.toLinear()</code> produces the formula in linear form:

<math>
\rho_3(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4) - b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3)
</math>

and <code>r3.toCode()</code> produces a version usable in Mathematica:

<pre>
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] + bp[x1, x3, x4]
- bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]
</pre>

===Syzygies===
<pre>
import syzygy;

// Phi around B
Braid initial;
initial.n=4;
initial.add(bp,2);
initial.add(bp,0);
initial.add(bp,1);
initial.add(bp,0);
initial.add(bp,2);
initial.add(phi,1);

Syzygy pb;
pb.lsym="\Phi B";
pb.codename="PhiAroundB";
pb.initial=initial;
pb.apply(r3,1,0);
pb.apply(r4a,3,1);
pb.swap(2,3);
pb.apply(r4b,0,1);
pb.apply(-r3,1,0);
pb.apply(-r4a,0,0);
pb.swap(2,3);
pb.apply(-r4b,3,0);
pb.apply(r3,1,1);

pb.draw();
</pre>

[[Image:06-1350-PhiAroundB.png|center]]

Again, like relations, we can use <code>pb.toLinear()</code>

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

and <code>pb.toCode()</code>

<pre>
PhiAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + rho4a[x1 + x5, x2, x3, x4]
+ rho4b[x1 + x2, x3, x4, x5] - rho3[x1, x2, x3 + x4, x5] - rho4a[x1, x2, x3, x4]
- rho4b[x1, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5]
</pre>

to produce formulas.

06-1350/Syzygies in Asymptote

2006-12-04T23:21:29Z

Andy: Added headings.

===Disclaimer===
These instructions (and the program they describe) are a work in progress and should be considered highly unreliable.

===Installation===
To use the syzygy script, first install [http://asymptote.sourceforge.net Asymptote]. Instructions for installing the program on several OSes is given in the documentation at the Asymptote website. The documentation also gives helpful instructions on how to run a script in Asymptote to produce a picture. Once installed, download [http://www.math.utoronto.ca/~andy/syzygy.asy syzygy.asy] and put it in a directory where Asymptote can find it. You should also have (or install) a variant of TeX on your system, such as MiKTeX, so that Asymptote can typeset labels.

===Braids===
Once installed, we can draw a braid in Asymptote:

<pre>
import syzygy; // Accesses the syzygy module.
Braid b; // Start a new braid.
b.n=3; // The braid has three strands.
// The strands are numbered left to right starting at 0.
b.add(bp,0); // Add a overcrossing component starting at strand 0,
// the leftmost strand.
b.add(bm,1); // Add an undercrossing starting at strand 1.
b.add(phi,0); // Add a trivalent vertex that merges strands 0 and 1.
// Strand 2 is now renumbered as strand 1.
b.draw(); // Draw the resulting braid.
</pre>

When saved into an asy file, say <code>mybraid.asy</code> and run with Asymptote, the result is a picture:

[[Image:06-1350-mybraid.png|center]]

===Relations===
====Drawing====
To define a relation, we first define two braids, and then stick them into a <code>Relation</code> structure. The below script generates an R3 relation.

<pre>
import syzygy; // Access the syzygy module.
Braid l; // Define the left hand side of the relation.
l.n=3; l.add(bp,0); l.add(bp,1); l.add(bp,0);
Braid r; // Define the right hand side of the relation.
r.n=3; r.add(bp,1); r.add(bp,0); r.add(bp,1);

Relation r3; // Define a relation.
r3.lsym="\rho_3"; // Give the relation a name for when it is written in functional form.
r3.codename="rho3"; // Give the relation a name to be used by Mathematica.
r3.lhs=l; r3.rhs=r;
r3.draw();
</pre>

When saved into an asy file and run, this draws the two sides of the relation. If TeX is installed, Asymptote will also put a lovely equals sign, typeset by TeX, between the two figures.

[[Image:06-1350-R3-asy.png|center]]

====Outputting Equations====
We can also get useful equations out of the relation. The method <code>r3.toFormula()</code> will produce a string that is the formula for the relation.

<pre>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</pre>

This string can be written out to the standard output by <code>write(r3.toFormula())</code>. It can be written to a file by <code>file f=output("filename.txt"); write(f, r3.toFormula())</code>. The string is formatted so it can be put into TeX or a wiki page using math mode:

<math>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</math>

The method <code>r3.toLinear()</code> produces the formula in linear form:

<math>
\rho_3(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4) - b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3)
</math>

and <code>r3.toCode()</code> produces a version of the relation that can be used in Mathematica:

<pre>
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] + bp[x1, x3, x4]
- bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]
</pre>

A few relations, such as <code>r3</code>, are already defined in <code>syzygy.asy</code> but more should be added.

====Applying====
Now that we have relations, we can apply them to bigger braids. Let's start with the braid in the <math>\Phi</math> around B syzygy:

<pre>
import syzygy;
Braid b;
b.n=4;
b.add(bp,2);
b.add(bp,0);
b.add(bp,1);
b.add(bp,0);
b.add(bp,2);
b.add(phi,1);
</pre>

[[Image:06-1350-pbstart.png|center]]

After skipping the lowest knot, we can apply R3 to the next three knots:

<pre>
Braid bb=apply(r3, b, 1, 0);
</pre>

here <code>apply(r, b, k, n)</code> means we are applying the relation <code>r</code> to the braid <code>b</code> at the place in the braid found by counting <code>k</code> components up from the bottom component and <code>n</code> strands in from the leftmost strand. <code>apply</code> does not modify the original braid, but returns the result of applying the relation (stored here as <code>bb</code>):

[[Image:06-1350-pbnext.png|center]]

This went from the left hand side of the relation to the right hand side. To apply a relation in reverse, simply prefix it by a minus sign. For example <code>apply(-r3, bb, 1, 0)</code> will yield a braid equivalent to our original. When applying a relation, the script first checks that the one side of the relation matches that portion of the braid, and will give a (somewhat cryptic) error if the relation cannot be applied.

In our braids, the components are placed from bottom to top in a fixed order. Sometimes when building syzygies, it is neccessary to swap the order that these components occur. This is done by the <code>swap</code> method. For instance, starting from <code>b</code>, we can swap the two bottom crossings:

<pre>
Braid swapped=b.swap(0,1);
</pre>

[[Image:06-1350-pbswap.png|center]]

Remember that components are ordered from bottom to top, starting at 0. Again, the script checks to make sure the swap is valid (ie. changing the order of the two components, doesn't actually change the knot) and will issue an error if it isn't.

===Syzygies===
One could manually apply relations and swaps, and make a whole bunch of braids, but it would be annoying to keep track of them all. Thankfully, the <code>Syzygy</code> structure does that for us. For example, here is the complete code for the <math>\Phi</math> around B syzygy:

<pre>
import syzygy;

// Phi around B
Braid initial;
initial.n=4;
initial.add(bp,2);
initial.add(bp,0);
initial.add(bp,1);
initial.add(bp,0);
initial.add(bp,2);
initial.add(phi,1);

Syzygy pb;
pb.lsym="\Phi B";
pb.codename="PhiAroundB";
pb.initial=initial;
pb.apply(r3,1,0);
pb.apply(r4a,3,1);
pb.swap(2,3);
pb.apply(r4b,0,1);
pb.apply(-r3,1,0);
pb.apply(-r4a,0,0);
pb.swap(2,3);
pb.apply(-r4b,3,0);
pb.apply(r3,1,1);

pb.draw();
</pre>

and the result

[[Image:06-1350-PhiAroundB.png|center]]

Again, like relations, we can use <code>pb.toLinear()</code>

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

and <code>pb.toCode()</code>

<pre>
PhiAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + rho4a[x1 + x5, x2, x3, x4]
+ rho4b[x1 + x2, x3, x4, x5] - rho3[x1, x2, x3 + x4, x5] - rho4a[x1, x2, x3, x4]
- rho4b[x1, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5]
</pre>

to give the formulas for the syzygies.

===Conclusion===
Example syzygyies can be found at my [[http://www.math.utoronto.ca/andy/ lame homepage]]. Please contact me if you have any questions or suggestions. Good luck and happy syzyging!

File:06-1350-pbnext.png

2006-12-04T23:00:46Z

Andy:

File:06-1350-pbswap.png

2006-12-04T23:00:02Z

Andy:

File:06-1350-pbstart.png

2006-12-04T22:59:35Z

Andy:

06-1350/Syzygies in Asymptote

2006-12-04T22:59:21Z

Andy:

WARNING: These instructions are a work in progress and should be considered highly unreliable.

To use the syzygy script, first install [http://asymptote.sourceforge.net Asymptote]. Instructions for installing the program on several OSes is given in the documentation at the Asymptote website. The documentation also gives helpful instructions on how to run a script in Asymptote to produce a picture. Once installed, download [http://www.math.utoronto.ca/~andy/syzygy.asy syzygy.asy] and put it in a directory where Asymptote can find it. You should also have (or install) a variant of TeX on your system, such as MiKTeX, so that Asymptote can typeset labels.

Once installed, we can draw a braid in Asymptote:

<pre>
import syzygy; // Accesses the syzygy module.
Braid b; // Start a new braid.
b.n=3; // The braid has three strands.
// The strands are numbered left to right starting at 0.
b.add(bp,0); // Add a overcrossing component starting at strand 0,
// the leftmost strand.
b.add(bm,1); // Add an undercrossing starting at strand 1.
b.add(phi,0); // Add a trivalent vertex that merges strands 0 and 1.
// Strand 2 is now renumbered as strand 1.
b.draw(); // Draw the resulting braid.
</pre>

When saved into an asy file, say <code>mybraid.asy</code> and run with Asymptote, the result is a picture:

[[Image:06-1350-mybraid.png|center]]

To define a relation, we first define two braids, and then stick them into a <code>Relation</code> structure. The below script generates an R3 relation.

<pre>
import syzygy; // Access the syzygy module.
Braid l; // Define the left hand side of the relation.
l.n=3; l.add(bp,0); l.add(bp,1); l.add(bp,0);
Braid r; // Define the right hand side of the relation.
r.n=3; r.add(bp,1); r.add(bp,0); r.add(bp,1);

Relation r3; // Define a relation.
r3.lsym="\rho_3"; // Give the relation a name for when it is written in functional form.
r3.codename="rho3"; // Give the relation a name to used by Mathematica.
r3.lhs=l; r3.rhs=r;
r3.draw();
</pre>

When saved into an asy file and run, this draws the two sides of the relation. If TeX is installed, Asymptote will also put a lovely equals sign, typeset by TeX, between the two figures.

[[Image:06-1350-R3-asy.png|center]]

We can also get useful equations out of the relation. The method <code>r3.toFormula()</code> will produce a string that is the formula for the relation.

<pre>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</pre>

This string can be written out to the standard output by <code>write(r3.toFormula())</code>. It can be written to a file by <code>file f=output("filename.txt"); write(f, r3.toFormula())</code>. The string is formatted so it can be put into TeX or a wiki page using math mode:

<math>
(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ =
(1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+
</math>

The method <code>r3.toLinear()</code> produces the formula in linear form:

<math>
\rho_3(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4) - b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3)
</math>

and <code>r3.toCode()</code> produces a version of the relation that can be used in Mathematica:

<pre>
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] + bp[x1, x3, x4]
- bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]
</pre>

A few relations, such as <code>r3</code>, are already defined in <code>syzygy.asy</code> but more should be added.

Now that we have relations, we can apply them to bigger braids. Let's start with the braid in the <math>\Phi</math> around B syzygy:

<pre>
import syzygy;
Braid b;
b.n=4;
b.add(bp,2);
b.add(bp,0);
b.add(bp,1);
b.add(bp,0);
b.add(bp,2);
b.add(phi,1);
</pre>

[[Image:06-1350-pbstart.png|center]]

After skipping the lowest knot, we can apply R3 to the next three knots:

<pre>
Braid bb=apply(r3, b, 1, 0);
</pre>

here <code>apply(r, b, k, n)</code> means we are applying the relation <code>r</code> to the braid <code>b</code> at the place in the braid found by counting <code>k</code> components up from the bottom component and <code>n</code> strands in from the leftmost strand. <code>apply</code> does not modify the original braid, but returns the result of applying the relation (stored here as <code>bb</code>):

[[Image:06-1350-pbnext.png|center]]

This went from the left hand side of the relation to the right hand side. To apply a relation in reverse, simply prefix it by a minus sign. For example <code>apply(-r3, bb, 1, 0)</code> will yield a braid equivalent to our original. When applying a relation, the script first checks that the one side of the relation matches that portion of the braid, and will give a (somewhat cryptic) error if the relation cannot be applied.

In our braids, the components are placed from bottom to top in a fixed order. Sometimes when building syzygies, it is neccessary to swap the order that these components occur. This is done by the <code>swap</code> method. For instance, starting from <code>b</code>, we can swap the two bottom crossings:

<pre>
Braid swapped=b.swap(0,1);
</pre>

[[Image:06-1350-pbswap.png|center]]

Remember that components are ordered from bottom to top, starting at 0. Again, the script checks to make sure the swap is valid (ie. changing the order of the two components, doesn't actually change the knot) and will issue an error if it isn't.

One could manually apply relations and swaps, and make a whole bunch of braids, but it would be annoying to keep track of them all. Thankfully, the <code>Syzygy</code> structure does that for us. For example, here is the complete code for the <math>\Phi</math> around B syzygy:

<pre>
import syzygy;

// Phi around B
Braid initial;
initial.n=4;
initial.add(bp,2);
initial.add(bp,0);
initial.add(bp,1);
initial.add(bp,0);
initial.add(bp,2);
initial.add(phi,1);

Syzygy pb;
pb.lsym="\Phi B";
pb.codename="PhiAroundB";
pb.initial=initial;
pb.apply(r3,1,0);
pb.apply(r4a,3,1);
pb.swap(2,3);
pb.apply(r4b,0,1);
pb.apply(-r3,1,0);
pb.apply(-r4a,0,0);
pb.swap(2,3);
pb.apply(-r4b,3,0);
pb.apply(r3,1,1);

pb.draw();
</pre>

and the result

[[Image:06-1350-PhiAroundB.png|center]]

Again, like relations, we can use <code>pb.toLinear()</code>

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

and <code>pb.toCode()</code>

<pre>
PhiAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + rho4a[x1 + x5, x2, x3, x4]
+ rho4b[x1 + x2, x3, x4, x5] - rho3[x1, x2, x3 + x4, x5] - rho4a[x1, x2, x3, x4]
- rho4b[x1, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5]
</pre>

to give the formulas for the syzygies.

File:06-1350-R3-asy.png

2006-12-04T22:58:22Z

Andy:

File:06-1350-mybraid.png

2006-12-04T22:57:15Z

Andy:

06-1350/Syzygies in Asymptote

2006-12-04T20:12:04Z

Andy: More write-up

06-1350/Syzygies in Asymptote

2006-12-04T19:45:57Z

Andy: Started guide.

User:Andy/06-1350-HW4

2006-12-03T23:20:22Z

Andy: Added equation for syzygy.

===The Generators===

Our generators are <math>T</math>, <math>R</math>, <math>\Phi</math> and <math>B^{\pm}</math>:
{| align=center cellpadding=10 style="border: solid orange 1px"
|- align=center valign=middle
|align=left|Picture
|
|
|
|[[Image:06-1350-BPlus.svg|100px]]
|
|- align=center valign=middle
|align=left|Generator
|<math>T</math>
|<math>R</math>
|<math>\Phi</math>
|<math>B^+</math>
|<math>B^-</math>
|- align=center valign=middle
|align=left|Perturbation
|<math>t</math>
|<math>r</math>
|<math>\varphi</math>
|<math>b^+</math>
|<math>b^-</math>
|}

===The Relations===

====The Reidemeister Move R3====
The picture (with three sides of the shielding removed) is
[[Image:06-1350-R4.svg|400px|center]]
In formulas, this is
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ = (1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_3(x_1, x_2, x_3, x_4) = </math>
|<math>b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4)</math>
|-
|
|<math>- b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3).</math>
|}

====The Reidemeister Move R4====
To establish the syzygy below, I needed two versions of R4. First:
[[Image:06-1350-R4a.png|center]]
In formulas, this is
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star \Phi = (1123)^\star \Phi (1233)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_{4a}(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + \phi(x_1,x_3,x_4) - \phi(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_3+x_4).</math>
|}

Second:
[[Image:06-1350-R4b.png|center]]
In formulas, this is
<center><math>(1123)^\star B^+ (1203)^\star B^+ (1231)^\star \Phi = (1230)^\star \Phi (1223)^\star B^+</math>.</center>
Linearized and written in functional form, this becomes
{| align=center
|-
|<math>\rho_{4b}(x_1,x_2,x_3,x_4) = b^+(x_1+x_2,x_3,x_4) + b^+(x_1,x_2,x_4) + \phi(x_1+x_4,x_2,x_3) - \phi(x_1,x_2,x_3) - b^+(x_1,x_2+x_3,x_4).</math>
|}

Are these independent, or can they be shown to be equivalent using other relations?

===The Syzygies===

====The "B around B" Syzygy====

The picture, with all shielding removed, is
{| align=center
|- align=center
|[[Image:06-1350-BAroundB.svg|center]]
|-
|align=right|(Drawn with [http://www.inkscape.org/ Inkscape])<br>(note that lower quality pictures are also acceptable)
|}

The functional form of this syzygy is

{| align=center
|-
|<math>BB(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1, x_2, x_3, x_5) + \rho_3(x_1 + x_5, x_2, x_3, x_4) - \rho_3(x_1 + x_2, x_3, x_4, x_5)</math>
|-
|
|<math>- \rho_3(x_1, x_2, x_4, x_5) - \rho_3(x_1 + x_4, x_2, x_3, x_5) - \rho_3(x_1, x_2, x_3, x_4)</math>
|-
|
|<math>+ \rho_3(x_1, x_3, x_4, x_5) + \rho_3(x_1 + x_3, x_2, x_4, x_5).</math>
|}

====The "<math>\Phi</math> around B" Syzygy====

The picture, with all shielding (and any other helpful notations) removed, is
{| align=center
|- align=center
|[[Image:06-1350-PhiAroundB.png|center]]
|-
|align=right|(Drawn with [http://asymptote.sf.net/ Asymptote])
|}

The functional form of this syzygy is

{| align=center
|-
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math>
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math>
|-
|
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math>
|-
|
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math>
|}

===A Mathematica Verification===

The following simulated Mathematica session proves that for our single relation and single syzygy, <math>d^2=0</math>. Copy paste it into a live Mathematica session to see that it's right!

{{In|n=1|in=<nowiki>d1 = {
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] +
bp[x1, x3, x4] - bp[x1 + x2, x3, x4] - bp[x1, x2, x4] -
bp[x1 + x4, x2, x3]
};
d2 = {
BAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] +
rho3[x1 + x5, x2, x3, x4] - rho3[x1 + x2, x3, x4, x5] -
rho3[x1, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5] -
rho3[x1, x2, x3, x4] + rho3[x1, x3, x4, x5] +
rho3[x1 + x3, x2, x4, x5]
};</nowiki>}}

{{InOut|n=3|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2</nowiki>|out=<nowiki>- rho3[x1, x2, x3, x4] + rho3[x1, x2, x3, x5] - rho3[x1, x2, x4, x5]
+ rho3[x1, x3, x4, x5] - rho3[x1 + x2, x3, x4, x5]
+ rho3[x1 + x3, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5]
+ rho3[x1 + x5, x2, x3, x4]</nowiki>}}

{{InOut|n=4|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1</nowiki>|out=<nowiki>0</nowiki>}}

File:06-1350-R4b.png

2006-12-03T23:12:47Z

Andy:

User:Andy/06-1350-HW4

2006-12-03T23:12:30Z

Andy: Adding R4 relations.

File:06-1350-R4a.png

2006-12-03T23:09:55Z

Andy:

File:R4b.png

2006-12-03T23:06:46Z

Andy:

File:R4a.png

2006-12-03T23:06:34Z

Andy:

User:Andy/06-1350-HW4

2006-12-03T17:11:31Z

Andy: Added Phi around B picture.

File:06-1350-PhiAroundB.png

2006-12-03T17:03:00Z

Andy: A minimal picture showing a Phi around B+ syzygy.

A minimal picture showing a Phi around B+ syzygy.

User:Andy/06-1350-HW4

2006-11-27T21:51:41Z

Andy: Copied the assignment.

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2006-11-08T15:43:18Z

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2006-11-08T15:40:27Z

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2006-11-08T15:39:41Z

Andy: Added scanned notes

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2006-11-08T15:38:14Z

Andy: Added scanned notes.

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File:06-1350-scan1102-0005.jpg

2006-11-08T15:33:11Z

Andy:

File:06-1350-scan1102-0004.jpg

2006-11-08T15:33:03Z

Andy:

File:06-1350-scan1102-0003.jpg

2006-11-08T15:32:56Z

Andy:

File:06-1350-scan1102-0002.jpg

2006-11-08T15:32:49Z

Andy:

File:06-1350-scan1102-0001.jpg

2006-11-08T15:32:41Z

Andy:

File:06-1350-scan1031-0003.jpg

2006-11-08T15:32:31Z

Andy:

File:06-1350-scan1031-0002.jpg

2006-11-08T15:32:23Z

Andy:

File:06-1350-scan1031-0001.jpg

2006-11-08T15:32:15Z

Andy: