View source for Finite Type (Vassiliev) Invariants

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{{Startup Note}}

<!--$$?Vassiliev$$-->
<!--Robot Land, no human edits to "END"-->
{{HelpLine|
n  = 2 |
in = <nowiki>Vassiliev</nowiki> |
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>}}
<!--END-->

Thus, for example, let us reproduce [http://www.ma.hw.ac.uk/~simon/fishing.html Willerton's "fish"] ({{arXiv|math.GT/0104061}}), the result of plotting the values of <math>V_2(K)</math> against the values of <math>\pm V_3(K)</math>, where <math>V_2(K)</math> is the (standardly normalized) type 2 invariant of <math>K</math>, <math>V_3(K)</math> is the (standardly normalized) type 3 invariant of <math>K</math>, and where <math>K</math> runs over a set of knots with equal  crossing numbers (10, in the example below):

<!--$$ListPlot[
  Join @@ Table[
    K = Knot[10, k] ; v2 = Vassiliev[2][K]; v3 = Vassiliev[3][K];
    {{v2, v3}, {v2, -v3}},
    {k, 165}
  ],
  PlotStyle -> PointSize[0.02], PlotRange -> All, AspectRatio -> 1
]$$-->
<!--Robot Land, no human edits to "END"-->
{{Graphics|
n  = 3 |
in = <nowiki>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
]</nowiki> |
img= Finite_Type_Vassiliev_Invariants_Out_3.gif |
out= <nowiki>-Graphics-</nowiki>}}
<!--END-->



As another example, let us consider the expansion of the [[The Jones Polynomial|Jones polynomial]] for a knot <math>K</math> as a power series in <math>x</math> when we substitute the standard variable <math>q</math> with <math>e^x</math> and use the power series expansion of <math>e^x</math>:
<center><math>J(K)(q=e^x)=\sum_{n=0}^\infty\ V_n(K)x^n</math></center>
Then, for the above coefficients we have that <math>V_0(K)=1</math> and for all <math>n\ge 1</math> <math>V_n</math> is a [http://en.wikipedia.org/wiki/Finite_type_invariant Vassiliev invariant] of type <math>n</math> {{ref|BirmanLin}}.
We can see this result by using the invariant formula:
<center><math>V\left(\doublepoint\right)= V\left(\overcrossing\right)-V\left(\undercrossing\right)</math></center>
to check the Birman-Lin condition, which tells us that an invariant <math>V</math> is of type <math>m</math> if it vanishes on knots with more than <math>m</math> [http://mathworld.wolfram.com/OrdinaryDoublePoint.html double points], or self intersections (see {{ref|Bar-Natan}}). Computing <math>V</math> on knots with more than one double point by resolving one self intersection at a time, it is enough to check that <math>V</math> vanishes on knots with <math>m+1</math> double points:
<center><math>V\underbrace{ \left(\doublepoint\cdots\doublepoint\right) }_{m+1}=0</math></center>

The following two programs let us determine <math>V_n(K)</math> for any integer <math>n</math> and knot <math>K</math>:<br />

<!--$$SetCrossing[K_, l_Integer, s_] := Module[
   {pd, n},
   pd = PD[K];
   If[PositiveQ[pd[[l]]],
    If[s == "-", pd[[l]] = RotateRight@pd[[l]]], 
    If[s == "+", pd[[l]] = RotateLeft@pd[[l]]]];
   pd];$$-->
<!--Robot Land, no human edits to "END"-->
{{In|
n  = 4 |
in = <nowiki>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];</nowiki>}}
<!--END--> 
<!--$$V[K_, n_] := Series[Jones[K][Exp[x]], {x, 0, n}];
V[K_, n_, {i1_, is___}] := 
 V[SetCrossing[K, i1, "+"], n, {is}] - 
  V[SetCrossing[K, i1, "-"], n, {is}]; 
V[K_, n_, {}] := V[K, n];$$-->
<!--Robot Land, no human edits to "END"-->
{{In|
n  = 5 |
in = <nowiki>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];</nowiki>}}
<!--END-->
The first program, <code>SetCrossing</code>, sets the <math>l^{th}</math> crossing of a knot <math>K</math> to be positive or negative depending on whether we choose <math>s</math> to be "<math>+</math>" or "<math>-</math>". The second program uses the invariant formula to give the series expansion of the Jones polynomial of a knot <math>K</math> discussed above, up to order <math>x^n</math>, where a selected list of the crossings of <math>K</math> are taken as double points. <math>V_n(K)</math> is then the coefficient of the term containing <math>x^n</math>.

For example, we can check that <math>V_4</math> disappears on the knot [[9_47]] with its first five crossings taken as double points:
<!--$$V[Knot[9, 47], 4, {1, 2, 3, 4, 5}]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n  = 6 |
in = <nowiki>V[Knot[9, 47], 4, {1, 2, 3, 4, 5}]</nowiki> |
out= <nowiki>V[Knot[9, 47], 4, {1, 2, 3, 4, 5}]</nowiki>}}
<!--END-->
[[Image:9_47_doublepoint.GIF|center|thumb|200px|The knot [[9_47]] with its first five crossings taken as double points.]]

{{note|Bar-Natan}}D. Bar-Natan, ''On the Vassiliev Knot Invariants'', Topology '''34''' (1995) 423-472.

{{note|BirmanLin}}J.S. Birman and X.-S. Lin, ''Knot Polynomials and Vassiliev's Invariants'', Invent. Math. '''111''' (1993) 225-270.