$p$-adic framed braids II

Details $p$-adic framed braids II $p$-adic framed braids II

2009-05-22

arxiv.org/abs/0905.3626v2

Mathematics

Geometric Topology

43 pages, 6 figures, added references for Introduction, corrected typos in Theorem 8

Scientific paper

In this paper, which is sequel to \cite{jula}, we explore further the structures of the $p$-adic framed braids and the $p$-adic Yokonuma-Hecke algebras constructed in \cite{jula}, by means of dense sub-structures approximating $p$-adic elements. We then construct a $p$-adic Markov trace on the $p$-adic Yokonuma-Hecke algebras, which arises naturally as the inverse limit of classical Markov traces constructed in \cite{ju}, and we approximate the values of the $p$-adic trace on $p$-adic elements. Surprisingly, the traces in \cite{ju} do not normalize directly to yield isotopy invariants of oriented framed links. This leads to imposing the `$E$-condition' to the trace parameters. For solutions of the `$E$-system' we then define ${\Bbb C}$-valued isotopy invariants of oriented framed links, which lift to isotopy invariants of $p$-adic framed links. The Yokonuma-Hecke algebras have topological interpretations in the context of framed braids, of singular braids and of classical braids.

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