An adelic extension of the Jones polynomial

TL;DR

This paper introduces an adelic extension of the Jones polynomial by constructing adelic Yokonuma--Hecke algebras, defining an adelic Markov trace, and deriving new knot invariants satisfying cubic skein relations.

Contribution

It develops the theory of adelic Yokonuma--Hecke algebras and constructs new knot invariants based on adelic Markov traces, extending previous p-adic approaches.

Findings

Defined the completion of the framed braid group

Constructed an adelic Markov trace for knot invariants

Produced invariants satisfying cubic skein relations

Abstract

In this paper we represent the classical braids in the Yokonuma--Hecke and the adelic Yokonuma--Hecke algebras. More precisely, we define the completion of the framed braid group and we introduce the adelic Yokonuma--Hecke algebras, in analogy to the $p$--adic framed braids and the $p$--adic Yokonuma--Hecke algebras introduced in \cite{jula,jula2}. We further construct an adelic Markov trace, analogous to the $p$--adic Markov trace constructed in \cite{jula2}, and using the traces in \cite{ju} and the adelic Markov trace we define topological invariants of classical knots and links, upon imposing some condition. Each invariant satisfies a cubic skein relation coming from the Yokonuma--Hecke algebra.