Markov traces on affine and cyclotomic Yokonuma-Hecke algebras

Abstract

In this article, we define and study the affine and cyclotomic Yokonuma-Hecke algebras. These algebras generalise at the same time the Ariki-Koike and affine Hecke algebras and the Yokonuma-Hecke algebras. We study the representation theory of these algebras and construct several bases for them. We then show how we can define Markov traces on them, which we in turn use to construct invariants for framed and classical knots in the solid torus. Finally, we study the Markov trace with zero parameters on the cyclotomic Yokonuma-Hecke algebras and determine the Schur elements with respect to that trace.