Enabling computations for link invariants coming from the Yokonuma–Hecke algebras

Abstract

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.

A new two-variable generalization of the Jones polynomial

“…Using the skein relation (1) of Theorem 3, it was proved in [4] that the invariants Θ d distinguish a pair of P -equivalent links (in fact more pairs were found in [4] using computational methods [24]). Since the invariant Θ contains the Homflypt polynomial as well as the family of invariants {Θ d }, one can now easily derive the following:…”

“…Eventually, in [4] the implementation of a different presentation for the algebra Y d,n , with parameter q instead of u, revealed that the corresponding invariants for classical links, Θ d (q, λ D ), satisfy the skein relation of the Homflypt polynomial P , but only on crossings between different components of the link, and this fact allowed the comparison of the invariants Θ d to P [24,6,4]. We note that for d = 1 the invariant Θ 1 coincides with P .…”