Narrow Results

In this paper, we study properties of the Markov trace tr${}_d$ and the specialized trace ${\mathrm{\text{tr}}}_{d,D}$ on the Yokonuma–Hecke algebras, such as behavior under inversion of a word, connected sums and mirror imaging. We then define invariants for framed, classical and singular links through the trace ${\mathrm{\text{tr}}}_{d,D}$ and also invariants for transverse links through the trace 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 Homflypt-equivalent pairs of knots and links. Our computations lead to the result that these invariants are topologically equivalent to the Homflypt polynomial on knots. However, they do not demonstrate the same behavior on links.

We describe an algorithm for computing the invariants of classical links arising from the Yokonuma–Hecke algebras. A detailed overview of the algorithm is given, following closely its implementation, a program used to calculate the invariants on several Homflypt-equivalent pairs of links.