p-Adic framed braids

Abstract

The Yokonuma-Hecke algebras are quotients of the modular framed braid group and they support Markov traces. In this paper, which is sequel to [6], we explore further the structures of the p-adic framed braids and the p-adic Yokonuma-Hecke algebras constructed in [6], by means of dense sub-structures approximating p-adic elements. We also construct a p-adic Markov trace on the p-adic Yokonuma-Hecke algebras and we approximate the values of the p-adic trace on p-adic elements. Surprisingly, the Markov traces do not re-scale directly to yield isotopy invariants of framed links. This leads to imposing the 'E-condition' on the trace parameters. For solutions of the 'E-system' we then define 2-variable isotopy invariants of modular framed links. These lift to p-adic isotopy invariants of classical framed links. The Yokonuma-Hecke algebras have topological interpretations in the context of framed knots, of classical knots of singular knots and of transverse knots.1991 Mathematics Subject Classification. 57M27, 20F38, 20F36, 20C08.

The Yokonuma–hecke Algebras and the Homflypt Polynomial

“…For d = 1, the algebra Y 1,n (u) coincides with the Iwahori-Hecke algebra H n (u) of type A. For more details and for further topological interpretations, see [JuLa1,JuLa2,JuLa3,JuLa4] Clearly e i,k = e k,i and it can be easily deduced that e 2 i,k = e i,k . Note that e i,i = 1 and that e i,i+1 = e i .…”

“…t kn n and the 'braiding part' σ. Applying further the braid relations (2.1)(b 1 ) and (2.1)(b 2 ) and the quadratic relations (2.2), we deduce that the following set is a C-basis for Y d,n (u) [Ju,JuLa1]:…”

An isomorphism theorem for Yokonuma–Hecke algebras and applications to link invariants

“…We note that we use a different definition of a Markov trace on Y d,n than in [10,12,13,14,15]. In there, the standard approach initiated by Jones for classical links was followed (see [8] and references therein).…”

“…This trace was subsequently used by Juyumaya and Lambropoulou to produce isotopy invariants for framed links [12,15]. Remarkably, they also produced isotopy invariants for classical links and singular links [13,14].…”

Determination of the Representations and a Basis for the Yokonuma–Temperley–Lieb Algebra

“…Thanks to the latter description, the Yokonuma-Hecke algebra has interesting topological interpretations in the context of framed knots and links. Juyumaya and Lambropoulou used Y d,n (u) to define knot invariants for framed knots [JuLa1,JuLa2]. They subsequently proved that these invariants can be extended to classical and singular knots [JuLa3,JuLa4].…”