Finite Type (Vassiliev) Invariants

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(For In[1] see Setup)

In[1]:= ?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-