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 [math]\displaystyle{ V_2(K) }[/math] against the values of [math]\displaystyle{ \pm V_3(K) }[/math], where [math]\displaystyle{ V_2(K) }[/math] is the (standardly normalized) type 2 invariant of [math]\displaystyle{ K }[/math], [math]\displaystyle{ V_3(K) }[/math] is the (standardly normalized) type 3 invariant of [math]\displaystyle{ K }[/math], and where [math]\displaystyle{ K }[/math] 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 ]
Out[3]=	-Graphics-

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

[math]\displaystyle{ J(K)(q=e^x)=\sum_{n=0}^\infty\ V_n(K)x^n }[/math]

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

[math]\displaystyle{ V\left(\doublepoint\right)= V\left(\overcrossing\right)-V\left(\undercrossing\right) }[/math]

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

[math]\displaystyle{ V\underbrace{ \left(\doublepoint\cdots\doublepoint\right) }_{m+1}=0 }[/math]

The following two programs let us determine [math]\displaystyle{ V_n(K) }[/math] for any integer [math]\displaystyle{ n }[/math] and knot [math]\displaystyle{ K }[/math]:

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

For example, we can check that [math]\displaystyle{ V_4 }[/math] 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.