An adelic extension of the Jones polynomial

Details An adelic extension of the Jones polynomial An adelic extension of the Jones polynomial

2009-09-14

arxiv.org/abs/0909.2545v2

1. M. Banagl, D. Vogel (eds.) The mathematics of knots, Contributions in the Mathematical and Computational Sciences, Vol. 1,

Mathematics

Geometric Topology

15 pages, 1 figure

Scientific paper

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.

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