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 ]
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 $n\geq 1$ $V_{n}$ is a Vassiliev invariant of type $n$ [BirmanLin]. We can see this result by using the invariant formula:

Failed to parse (unknown function "\doublepoint"): {\displaystyle 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:

Failed to parse (unknown function "\doublepoint"): {\displaystyle 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.