06-1350/Syzygies in Asymptote in Brief - Revision history

Drorbn at 01:54, 5 December 2006

2006-12-05T01:54:22Z

← Older revision		Revision as of 21:54, 4 December 2006
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	{{06-1350/Navigation}}

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

Drorbn at 01:53, 5 December 2006

2006-12-05T01:53:41Z

← Older revision		Revision as of 21:53, 4 December 2006
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			{{06-1350/Navigation}}

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

74.99.178.225: Formatting.

2006-12-05T01:47:16Z

Formatting.

← Older revision		Revision as of 21:47, 4 December 2006
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	===Braids===		===Braids===
	{\|align=center ~~cellpadding="5"~~		{\|align=center
	\|-		\|-
	\|<pre>		\|<pre>
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	b.draw(); // Draw the resulting braid.		b.draw(); // Draw the resulting braid.
	</pre>		</pre>
			\|width="40pt"\|
	\|[[Image:06-1350-mybraid.png]]		\|[[Image:06-1350-mybraid.png]]
	\|}		\|}

	===Relations===		===Relations===
	{\|align=center ~~cellpadding="5"~~		{\|align=center
	\|-		\|-
	\|<pre>		\|<pre>
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	r3.draw();		r3.draw();
	</pre>		</pre>
			\|width="40pt"\|
	\|[[Image:06-1350-R3-asy.png]]		\|[[Image:06-1350-R3-asy.png]]
	\|}		\|}
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	===Syzygies===		===Syzygies===

	{\|align=center ~~cellpadding="5"~~		{\|align=center
	\|-		\|-
	\|<pre>		\|<pre>
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	pb.draw();		pb.draw();
	</pre>		</pre>
			\|width="60pt"\|
	\|[[Image:06-1350-bpsmall.png]]		\|[[Image:06-1350-bpsmall.png]]
	\|}		\|}

Andy at 00:29, 5 December 2006

2006-12-05T00:29:34Z

← Older revision		Revision as of 20:29, 4 December 2006
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	<code>r3.toFormula()</code> produces the formula:		<code>r3.toFormula()</code> produces the formula:
		⚫	<center><math>

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

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

			{\|align=center
	<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>\rho_3(x_1,x_2,x_3,x_4) = </math>
	</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:		and <code>r3.toCode()</code> produces a version usable in Mathematica:

Andy at 00:25, 5 December 2006

2006-12-05T00:25:11Z

← Older revision		Revision as of 20:25, 4 December 2006
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	===Braids===		===Braids===
	{\|align=center cellpadding="20"		{\|align=center cellpadding="5"
	\|-		\|-
	\|<pre>		\|<pre>
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	===Relations===		===Relations===
	{\|align=center cellpadding="20"		{\|align=center cellpadding="5"
	\|-		\|-
	\|<pre>		\|<pre>
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	===Syzygies===		===Syzygies===

	{\|align=center cellpadding="20"		{\|align=center cellpadding="5"
	\|-		\|-
	\|<pre>		\|<pre>

Andy: Formatting.

2006-12-05T00:23:33Z

Formatting.

← Older revision		Revision as of 20:23, 4 December 2006
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	===Braids===		===Braids===
			{\|align=center cellpadding="20"
⚫	[[Image:06-1350-mybraid.png~~\|frame~~]]
			\|-
	<pre>		\|<pre>
	import syzygy; // Accesses the syzygy module.		import syzygy; // Accesses the syzygy module.
	Braid b; // Start a new braid.		Braid b; // Start a new braid.
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	b.draw(); // Draw the resulting braid.		b.draw(); // Draw the resulting braid.
	</pre>		</pre>
		⚫	\|[[Image:06-1350-mybraid.png]]
			\|}

	===Relations===		===Relations===
			{\|align=center cellpadding="20"
⚫	[[Image:06-1350-R3-asy.png~~\|frame~~]]
			\|-
	<pre>		\|<pre>
	import syzygy; // Access the syzygy module.		import syzygy; // Access the syzygy module.
	Braid l; // Define the left hand side of the relation.		Braid l; // Define the left hand side of the relation.
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	Relation r3; // Define a relation.		Relation r3; // Define a relation.
	r3.lsym="\rho_3"; // Give the relation a name ~~for when it is written in functional form~~.		r3.lsym="\rho_3"; // Give the relation a formula name.
	r3.codename="rho3"; // Give the relation a name to be used by Mathematica.		r3.codename="rho3"; // Give the relation a name to be used by Mathematica.
	r3.lhs=l; r3.rhs=r;		r3.lhs=l; r3.rhs=r;
	r3.draw();		r3.draw();
	</pre>		</pre>
		⚫	\|[[Image:06-1350-R3-asy.png]]

			\|}


	<code>r3.toFormula()</code> produces the formula:		<code>r3.toFormula()</code> produces the formula:
Line 59:		Line 63:
	===Syzygies===		===Syzygies===

			{\|align=center cellpadding="20"
⚫	[[Image:06-1350-bpsmall.png~~\|frame~~]]
			\|-
	<pre>		\|<pre>
	import syzygy;		import syzygy;

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

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

Andy at 23:48, 4 December 2006

2006-12-04T23:48:05Z

← Older revision		Revision as of 19:48, 4 December 2006
Line 5:		Line 5:

	===Braids===		===Braids===
		⚫	[[Image:06-1350-mybraid.png\|frame]]
	<pre>		<pre>
	import syzygy; // Accesses the syzygy module.		import syzygy; // Accesses the syzygy module.
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	b.draw(); // Draw the resulting braid.		b.draw(); // Draw the resulting braid.
	</pre>		</pre>

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

	===Relations===		===Relations===
		⚫	[[Image:06-1350-R3-asy.png\|frame]]
	<pre>		<pre>
	import syzygy; // Access the syzygy module.		import syzygy; // Access the syzygy module.
Line 35:		Line 35:
	</pre>		</pre>


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

	<code>r3.toFormula()</code> produces the formula:		<code>r3.toFormula()</code> produces the formula:
Line 58:		Line 58:

	===Syzygies===		===Syzygies===

		⚫	[[Image:06-1350-bpsmall.png\|frame]]
	<pre>		<pre>
	import syzygy;		import syzygy;
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	pb.draw();		pb.draw();
	</pre>		</pre>

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

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

Andy: /* Installation */

2006-12-04T23:37:01Z

Installation

← Older revision		Revision as of 19:37, 4 December 2006
Line 1:		Line 1:
	===Installation===		===Installation===
	See [[Syzygies in Asymptote]] for more detailed information.		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].		First install [http://asymptote.sourceforge.net Asymptote]. Once installed, download [http://www.math.utoronto.ca/~andy/syzygy.asy syzygy.asy].

Andy: For Handout.

2006-12-04T23:36:32Z

For Handout.

New page

===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.