## Vassiliev invariants

A knot invariant is said to be a Vassiliev invariant (or a finite type invariant) of order (or degree) $n$ if its extension vanishes on all singular knots with more than $n$ double points. A Vassiliev invariant is said to be of order (degree) $n$ if it is of order $\leq n$ but not of order $\leq n-1$.

Vassiliev invariants can be obtained as knot dependent coefficients in a loop expansion of the colored HOMFLY polynomial: $$\nonumber \mathcal{H}^{\mathcal{K}}_{R}=\sum\limits_{n=0}^{\infty}\hbar^n\sum\limits_{|\Delta|\leq n}\mathcal{C}_{\Delta}^{R}\sum_{m=0}^{n-|\Delta|} \left(v^{\mathcal{K}}_{\Delta,m}\right)_n\, N^m\,,$$ where $\mathcal{C}_{\Delta}^{R}$ are eigenvalues of $\mathfrak{sl}_N$ Casimir invariants, but not all of them are included in the basis group factors. In the list one uses the function Z[n_] for $\mathcal{C}_{[2k]}^{R}$ and the function AntiY[n_] for $\mathcal{C}_{[2k+1]}^{R}$. Also note that one should use the functions Z[2n1+1,2n2+1] for $\mathcal{C}_{[2n_1+1,2n_2+1]}^{R}$.

We label the non-zero Vassiliev invariants $\left(v^{\mathcal{K}}_{\Delta,m}\right)_n$ in lexicographical order as $v_{n,m}$. For example, for the 7-th level we have: $$\begin{array}{rl} v_{7,1}&:=\left(v_{[2],1}^\mathcal{K}\right)_7\,, \quad v_{7,2}&:=\left(v_{[2],3}^\mathcal{K}\right)_7\,, \quad v_{7,3}&:=\left(v_{[2],5}^\mathcal{K}\right)_7\,, \quad v_{7,4}&:=\left(v_{[2,2],1}^\mathcal{K}\right)_7\,, \\ \\ v_{7,5}&:=\left(v_{[4],1}^\mathcal{K}\right)_7\,, \quad v_{7,6}&:=\left(v_{[4],3}^\mathcal{K}\right)_7\,, \quad v_{7,7}&:=\left(v_{[6],1}^\mathcal{K}\right)_7\,, \quad v_{7,8}&:=\left(v_{[3,3],0}^\mathcal{K}\right)_7\,. \end{array}$$ List the primary Vassiliev invariants for the knot $\mathcal{K}=5_2$:

n \ m 123456789101112131415161718192021222324
2-4
3-6
4-1344
5-7-34222
6-701141/12186875-1684
7739/125621/125681/201623363421-17586-262
87135/615991/6321493/3605099/2-68419786159899/12-2322-34830-1119731354047208/3
9494347/3601965545/144559629/404931593/1680437326647/3172052/3421633/31046177/2047332-614212-613239240382217404/349187/319444
10164207/48813947/722527487/36202898261/201601531057/120131927/9-7590017822/94711697/65386709/615012799/72181479/22379224/31128632365745/2-651995037699381/1212942826278622681835-18664084307084/3889636/9781648/3

More Vassiliev invariants (for knots $3_1$, $5_2$).