# 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.

## Key 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.