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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in = <nowiki>Vassiliev</nowiki> | |
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| ⚫ | out= <nowiki>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.</nowiki>}} |
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<pre style="color: red; border: 0px; padding: 0em"><nowiki>ListPlot[ |
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in = <nowiki>ListPlot[ |
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Join @@ Table[ |
Join @@ Table[ |
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K = Knot[10, k] ; v2 = Vassiliev[2][K]; v3 = Vassiliev[3][K]; |
K = Knot[10, k] ; v2 = Vassiliev[2][K]; v3 = Vassiliev[3][K]; |
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PlotStyle -> PointSize[0.02], PlotRange -> All, AspectRatio -> 1 |
PlotStyle -> PointSize[0.02], PlotRange -> All, AspectRatio -> 1 |
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img= Finite_Type_Vassiliev_Invariants_Out_2.gif | |
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out= <nowiki>-Graphics-</nowiki>}} |
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Revision as of 14:19, 30 August 2005
(For In[1] see Setup)
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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[3]:=
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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
]
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Out[3]=
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-Graphics-
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