Finite Type (Vassiliev) Invariants

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(For In[1] see Setup)

In[2]:= ?Vassiliev
Vassiliev[2][K] computes the (standardly normalized) type 2 Vassiliev invariant of the knot K, i.e., the coefficient of z^2 in Conway[K][z]. Vassiliev[3][K] computes the (standardly normalized) type 3 Vassiliev invariant of the knot K, i.e., 3J''(1)-(1/36)J'''(1) where J is the Jones polynomial of K.

Thus, for example, let us reproduce Willerton's "fish" (arXiv:math.GT/0104061), the result of plotting the values of V_2(K) against the values of \pm V_3(K), where V_2(K) is the (standardly normalized) type 2 invariant of K, V_3(K) is the (standardly normalized) type 3 invariant of K, and where K runs over a set of knots with equal crossing numbers (10, in the example below):

In[3]:= ListPlot[ Join @@ Table[ K = Knot[10, k] ; v2 = Vassiliev[2][K]; v3 = Vassiliev[3][K]; {{v2, v3}, {v2, -v3}}, {k, 165} ], PlotStyle -> PointSize[0.02], PlotRange -> All, AspectRatio -> 1 ]
Finite Type Vassiliev Invariants Out 3.gif
Out[3]= -Graphics-

As another example, let us consider the expansion of the Jones polynomial for a knot K as a power series in x when we substitute the standard variable q with e^x and use the power series expansion of e^x:

J(K)(q=e^x)=\sum_{n=0}^\infty\ V_n(K)x^n

Then, for the above coefficients we have that V_0(K)=1 and for all n\ge 1 V_n is a Vassiliev invariant of type n [BirmanLin]. We can see this result by using the invariant formula:

V\left(\doublepoint\right)= V\left(\overcrossing\right)-V\left(\undercrossing\right)

to check the Birman-Lin condition, which tells us that an invariant V is of type m if it vanishes on knots with more than m double points, or self intersections (see [Bar-Natan]). Computing V on knots with more than one double point by resolving one self intersection at a time, it is enough to check that V vanishes on knots with m+1 double points:

V\underbrace{ \left(\doublepoint\cdots\doublepoint\right) }_{m+1}=0

The following two programs let us determine V_n(K) for any integer n and knot K:

In[4]:= SetCrossing[K_, l_Integer, s_] := Module[ {pd, n}, pd = PD[K]; If[PositiveQ[pd[[l]]], If[s == "-", pd[[l]] = RotateRight@pd[[l]]], If[s == "+", pd[[l]] = RotateLeft@pd[[l]]]]; pd];
In[5]:= V[K_, n_] := Series[Jones[K][Exp[x]], {x, 0, n}]; V[K_, n_, {i1_, is___}] := V[SetCrossing[K, i1, "+"], n, {is}] - V[SetCrossing[K, i1, "-"], n, {is}]; V[K_, n_, {}] := V[K, n];

The first program, SetCrossing, sets the l^{th} crossing of a knot K to be positive or negative depending on whether we choose s to be "+" or "-". The second program uses the invariant formula to give the series expansion of the Jones polynomial of a knot K discussed above, up to order x^n, where a selected list of the crossings of K are taken as double points. V_n(K) is then the coefficient of the term containing x^n.

For example, we can check that V_4 disappears on the knot 9_47 with its first five crossings taken as double points:

In[6]:= V[Knot[9, 47], 4, {1, 2, 3, 4, 5}]
Out[6]= V[Knot[9, 47], 4, {1, 2, 3, 4, 5}]
The knot 9_47 with its first five crossings taken as double points.

[Bar-Natan] ^ D. Bar-Natan, On the Vassiliev Knot Invariants, Topology 34 (1995) 423-472.

[BirmanLin] ^ J.S. Birman and X.-S. Lin, Knot Polynomials and Vassiliev's Invariants, Invent. Math. 111 (1993) 225-270.