Identifying the invariants for classical knots and links from the Yokonuma-Hecke algebras

Maria Chlouveraki, Jesus Juyumaya, Konstantinos Karvounis, Sofia Lambropoulou

Published 2015-05-25Version 1

In this paper we study the Juyumaya-Lambropoulou invariants $\Delta_{d,D}$ for classical knots and links constructed from the Yokonuma-Hecke algebras, and in particular their relationship to the Homflypt polynomial. We first prove that the Markov trace ${\rm tr}_{d,D}$ on the Yokonuma-Hecke algebras can be computed on classical knots and links by five rules which do not involve the framing generators. Using this we show that the invariants $\Delta_{d,D}$ on classical knots are topologically equivalent to the Homflypt polynomial. We further describe the behaviour of the invariants $\Delta_{d,D}$ on arbitrary links in relation to the Homflypt polynomial by means of a closed formula. Finally, we show that the invariants $\Delta_{d,D}$ are not topologically equivalent to the Homflypt polynomial by providing computational data for six pairs of Homflypt-equivalent links which are distinguished by $\Delta_{d,D}$ and a diagrammatic proof for one of these pairs.