Identifying the Invariants for Classical Knots and Links from the Yokonuma–Hecke Algebras

Author:

Chlouveraki Maria1,Juyumaya Jesús2,Karvounis Konstantinos3,Lambropoulou Sofia4

Affiliation:

1. Laboratoire de Mathématiques, UVSQ, Bâtiment Fermat, Versailles Cedex, France

2. Instituto de Matemáticas, Universidad de Valparaíso, Valparaíso, Chile

3. Institut für Mathematik, Universität Zürich, Zürich, Switzerland

4. Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens, Greece

Abstract

Abstract We announce the existence of a family of new 2-variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma–Hecke algebra of type A. Yokonuma–Hecke algebras are generalizations of Iwahori–Hecke algebras, and this family contains the HOMFLYPT polynomial, the famous 2-variable invariant for classical links arising from the Iwahori–Hecke algebra of type A. We show that these invariants are topologically equivalent to the HOMFLYPT polynomial on knots, but not on links, by providing pairs of HOMFLYPT-equivalent links that are distinguished by our invariants. In order to do this, we prove that our invariants can be defined diagrammatically via a special skein relation involving only crossings between different components. We further generalize this family of invariants to a new 3-variable skein link invariant that is stronger than the HOMFLYPT polynomial. Finally, we present a closed formula for this invariant, by W. B. R. Lickorish, that uses HOMFLYPT polynomials of sublinks and linking numbers of a given oriented link.

Funder

European Social Fund

National Strategic Reference Framework

Fondecyt

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

Reference38 articles.

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2. Tied links;Aicardi,2016

3. LinkInfo: Table of Knot Invariants;Cha,2015

4. “The Yokonuma-Hecke algebras and the HOMFLYPT polynomial;Chlouveraki,2013):

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