# Finite Type (Vassiliev) Invariants

(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 ${\displaystyle V_{2}(K)}$ against the values of ${\displaystyle \pm V_{3}(K)}$, where ${\displaystyle V_{2}(K)}$ is the (standardly normalized) type 2 invariant of ${\displaystyle K}$, ${\displaystyle V_{3}(K)}$ is the (standardly normalized) type 3 invariant of ${\displaystyle K}$, and where ${\displaystyle K}$ 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 ${\displaystyle K}$ as a power series in ${\displaystyle x}$ when we substitute the standard variable ${\displaystyle q}$ with ${\displaystyle e^{x}}$ and use the power series expansion of ${\displaystyle e^{x}}$:

${\displaystyle J(K)(q=e^{x})=\sum _{n=0}^{\infty }\ V_{n}(K)x^{n}}$

Then, for the above coefficients we have that ${\displaystyle V_{0}(K)=1}$ and for all ${\displaystyle n\geq 1}$ ${\displaystyle V_{n}}$ is a Vassiliev invariant of type ${\displaystyle n}$ . We can see this result by using the invariant formula:

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

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

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

The following two programs let us determine ${\displaystyle V_{n}(K)}$ for any integer ${\displaystyle n}$ and knot ${\displaystyle K}$:

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

For example, we can check that ${\displaystyle V_{4}}$ 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.