Finite Type (Vassiliev) Invariants
(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):
As another example, let us consider the expansion of the Jones polynomial for a knot as a power series in
when we substitute the standard variable
with
and use the power series expansion of
:

Then, for the above coefficients we have that and for all
is a Vassiliev invariant of type
[BirmanLin].
We can see this result by using the invariant formula:

to check the Birman-Lin condition, which tells us that an invariant is of type
if it vanishes on knots with more than
double points, or self intersections (see [Bar-Natan]). Computing
on knots with more than one double point by resolving one self intersection at a time, it is enough to check that
vanishes on knots with
double points:

The following two programs let us determine for any integer
and knot
:
In[4]:=
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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]:=
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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 crossing of a knot
to be positive or negative depending on whether we choose
to be "
" or "
". The second program uses the invariant formula to give the series expansion of the Jones polynomial of a knot
discussed above, up to order
, where a selected list of the crossings of
are taken as double points.
is then the coefficient of the term containing
.
For example, we can check that disappears on the knot 9_47 with its first five crossings taken as double points:
In[6]:=
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V[Knot[9, 47], 4, {1, 2, 3, 4, 5}]
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Out[6]=
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V[Knot[9, 47], 4, {1, 2, 3, 4, 5}]
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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.