Markov trace on Funar algebra

Abstract

Funar algebra $K_\infty=K_\infty(\alpha,\beta;k)$ is the quotient of the group algebra over a ring $k$ of the braid group $B_\infty$ by two cubic relations: $\sigma_1^3-\alpha\sigma_1^2+\beta\sigma_1-1=0$ and another one which involves $\sigma_1$ and $\sigma_2$. The universal Markov trace on $K_\infty$ is the quotient map $t$ of $K_\infty(\alpha,\beta,k[u,v])$ to its quotient (as a $k[u,v]$-module) by trace relations $xy=yx$ and by Markov relations $\sigma_nx=ux$, $\sigma_n^{-1}x=vx$ for $x\in K_n$. It is easy to check that the quotient is of the form $k[u,v]/I$ for some ideal $I$ (i. e. that the trace $t$ is determined by $t(1)$). We give an algorithm to compute the ideal $I$ and we present the result of computations in some special cases. In the last section we discuss some properties of the resulting link invariant. This invariant for $\beta=0$, $k=GF(37)[\alpha]$ detects the chirality of the knots $10_{48}$ and $10_{91}$ and it distinguish many other pairs of knots with equal HOMFLY polynomials.