Finite Type (Vassiliev) Invariants
From Knot Atlas
(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 V2(K) against the values of 
, where V2(K) is the (standardly normalized) type 2 invariant of K, V3(K) is the (standardly normalized) type 3 invariant of K, and where 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
]
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Out[3]=
| -Graphics-
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As another example, let us consider the expansion of the Jones polynomial for a knot K as a power series in x when we substitute the standard variable q with ex and use the power series expansion of ex:
Then, for the above coefficients we have that V0(K) = 1 and for all 
Vn is a Vassiliev invariant of type n [BirmanLin].
We can see this result by using the invariant formula:
to check the Birman-Lin condition, which tells us that an invariant V is of type m if it vanishes on knots with more than m double points, or self intersections (see [Bar-Natan]). Computing V on knots with more than one double point by resolving one self intersection at a time, it is enough to check that V vanishes on knots with m + 1 double points:
The following two programs let us determine Vn(K) for any integer n and knot 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];
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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];
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The first program, SetCrossing, sets the lth crossing of a knot K to be positive or negative depending on whether we choose s to be " + " or "−". The second program uses the invariant formula to give the series expansion of the Jones polynomial of a knot K discussed above, up to order xn, where a selected list of the crossings of K are taken as double points. Vn(K) is then the coefficient of the term containing xn.
For example, we can check that V4 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}]
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Out[6]=
| 5
O[x]
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[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.
