Finite Type (Vassiliev) Invariants: Difference between revisions
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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. |
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K = Knot[10, k] ; v2 = Vassiliev[2][K]; v3 = Vassiliev[3][K]; |
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{{v2, v3}, {v2, -v3}}, |
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{k, 165} |
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], |
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PlotStyle -> PointSize[0.02], PlotRange -> All, AspectRatio -> 1] |
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{{Graphics2|n=2|imagename=Finite_Type_%28Vassiliev%29_Invariants_Out_2.gif}} |
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Revision as of 22:43, 28 August 2005
(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 against the values of , where is the (standardly normalized) type 2 invariant of , is the (standardly normalized) type 3 invariant of , and where runs over a set of knots with equal crossing numbers (10, in the example below):
In[2]:= |
Show[ot[ 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] |
File:Finite Type (Vassiliev) Invariants Out 2.gif | |
Out[2]= | -Graphics- |