On the knot invariants from the Yokonuma-Hecke algebras

Sergei Chmutov, Slavik Jablan, Konstantinos Karvounis, Sofia Lambropoulou

Published 2015-05-25Version 1

In this paper we study properties of the Juyumaya trace ${\rm tr}_d$ and the specialized trace ${\rm tr}_{d,D}$ on the Yokonuma-Hecke algebras; namely behaviour under inversion of a word, connected sums and mirror imaging. We then review the Juyumaya-Lambropoulou invariants for framed, classical and singular links constructed through the trace ${\rm tr}_{d,D}$ and we also construct invariants for transverse links through the trace ${\rm tr}_d$. In order to compare the invariants for classical links with the Homflypt polynomial we develop computer programs and we evaluate them on several pairs of knots and links which are Homflypt-equivalent. Our computations lead to our conjecture that these invariants are topologically equivalent to the Homflypt polynomial on knots. However, they do not demonstrate the same behaviour on links.