VasCalc Documentation - An example

This is an example of how to use VasCalc. We check the third Reidemeister move against an almost-invariant (not a technical term).

The Reidemeister 3 Move

We want to use VasCalc to verify the third Reidemeister move. This is meant as a small example of how to use the VasCalc package.

The first couple of steps are to load up VasCalc.

In[1]:=

<<CDinterface.m

In[2]:=

SetVasCalcPath["/home/zavosh/vc"];

Now we need to load the definitions of $\Phi$ and $R$ as defined in Dror's paper on Non-Associative Tangles:

In[3]:=	Phi = ASeries[1 + (1/24)*CD[Line[1], Line[2], Line[1, 2]] - (1/24)*CD[Line[2], Line[1], Line[1, 2]], 3, 0, 3]
Out[3]=	ASeries[3, 0, {«JavaObject[vectorSpace.Coefficient]», «JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]», «JavaObject[vectorSpace.Coefficient]»}]

In[4]:=	R = ASeries[1 + (1/2)CD[Line[1], Line[1]] + (1/8)CD[Line[1, 2], Line[1, 2]] + (1/48)CD[Line[1, 2, 3], Line[1, 2, 3]] , 2, 0]
Out[4]=	ASeries[2, 0, {«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]», «JavaObject[ChordVector]», «JavaObject[ChordVector]»}]

Now we define the definition of Z(B(k),n). Where Z stands for our (pseudo) Vassiliev invariant and Z(B(k),n) is the invariant of a braid on n strands with a crossing on the strands k and k+1.

In[5]:=

Clear[Z]

In[6]:=

Z[B[k_],n_] /; k > 1 := Module[{ser, tbl}, ser = Nest[AddStrand[#, #[[1]]] &, Nest[DoubleStrand[#, 0] &, Phi, k - 2], n - k - 1]. Nest[AddStrand[#, #[[1]]] &, Nest[AddStrand[#, 0] &, R, k - 1], n - k - 1] . Nest[AddStrand[#, #[[1]]] &, Nest[DoubleStrand[#, 0] &, PermuteStrand[((Phi))^(-1), {{2, 3}}], k - 2], n - k - 1]; tbl = Table[i, {i, ser[[1]]}]; tbl = ReplacePart[tbl, k + 1, k]; tbl = ReplacePart[tbl, k, k + 1]; ASeries[tbl, ser[[3]]]]