User:Karenechu/06-1350-HW4

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The Generators

Our generators are , , and :

Picture 06-1350-BPlus.svg
Generator
Perturbation

A low-tech completed version of this chart (Suzie's):

Chart.jpg

The Relations

The Symmetry of B (Suzie's)

To eliminate the choice involved in placing a B at a crossing, it has to have 180 degrees rotational symmetry. This yields the following picture:

Symm1.jpg

The relation cannot be written in the first notation, as on the right side the chords ending on different red lines could end up on the same pink line.

In the linearized functional notation though we can express this:

Explanation: on the right side, chords on the first red line can drop off on either the third, second or the first strand, morover, the orders are reversed, hence the minus signs.

The same picture for B^- yields:

The symmetry of (Suzie's)

has to have A(4)-symmetry. For example, rotation around the "top" vertex yields the following picture and relation:

Symm3.jpg

The same explanation goes here, and we get the relation:

And here is the full picture and the relations (I suppose it would be enough to take a few that generate A(4), but we're on the safe side writing all these up... and I kind of got into drawing tetrahedrons.):

Tetrahedrons.jpg

The Reidemeister move R1 (Suzie's)

Reidemeister1.jpg

As with the symmetry relations, we cannot write this one in the first notation either.

In the linearized functional notation, it looks like this:

Where the negative sign is because the order of the chords is reversed as we slide them along the little loop.

The Reidemeister move R2 (Suzie's)

With three sides of the shielding removed, the picture is: Reidemeister2.jpg

This means:

Linearized and in functional form:

And we get the other R2 by switching both crossings, i.e. switching b^+ and b^-:


The Reidemeister Move R3

The picture (with three sides of the shielding removed) is

06-1350-R4.svg

In formulas, this is

.

Linearized and written in functional form, this becomes

The Reidemeister Move R4, source:Andy

First version of R4:

06-1350-R4a.png

In formulas, this is

.

Linearized and written in functional form, this becomes

Second version:

06-1350-R4b.png

In formulas, this is

.

Linearized and written in functional form, this becomes


The Operation unzip

Warning: I am not sure if this is correct, but thought it better to post than not post.

We have found the Failed to parse (syntax error): {\displaystyle u^A,d^A,#^A} operations in the space corresponding to the Failed to parse (syntax error): {\displaystyle u, d, #} , in the space of , so that for the instance


Unzip.jpg


This means:

Linearized and in functional form:

And we get the other R2 by switching both crossings, i.e. switching b^+ and b^-:

The Syzygies

The "B around B" Syzygy

The picture, with all shielding removed, is

06-1350-BAroundB.svg
(Drawn with Inkscape)
(note that lower quality pictures are also acceptable)

The functional form of this syzygy is

The " around B" Syzygy- I copy-pasted this from Andy, as well as R4

The picture, with all shielding (and any other helpful notations) removed, is

06-1350-PhiAroundB.png
(Drawn with Asymptote, Syzygies in Asymptote)

The functional form of this syzygy is

The " around " Syzygy -also taken from Andy

note: I've changed Andy's notation to fit my version of R2.

The picture is

06-1350-PhiAroundPhi.png
(Drawn with Asymptote, Syzygies in Asymptote)

The functional form of this syzygy is

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, . Copy paste it into a live Mathematica session to see that it's right!

In[1]:= 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] };
In[3]:= BAroundB[x1, x2, x3, x4, x5] /. d2
Out[3]= - 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]
In[4]:= BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1
Out[4]= 0