  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.

Geometric Topology;Quantum Algebra