  The Yokonuma-Hecke algebras are quotients of the modular framed braid group
and they support Markov traces. In this paper, which is sequel to Juyumaya and
Lambropoulou (2007), we explore further the structures of the $p$-adic framed
braids and the $p$-adic Yokonuma-Hecke algebras constructed in Juyumaya and
Lambropoulou (2007), by means of dense sub-structures approximating $p$-adic
elements. We also construct a $p$-adic Markov trace on the $p$-adic
Yokonuma-Hecke algebras and we approximate the values of the $p$-adic trace on
$p$-adic elements. Surprisingly, the Markov traces do not re-scale directly to
yield isotopy invariants of framed links. This leads to imposing the
`$E$-condition' on the trace parameters. For solutions of the `$E$-system' we
then define 2-variable isotopy invariants of modular framed links. These lift
to $p$-adic isotopy invariants of classical framed links. The Yokonuma-Hecke
algebras have topological interpretations in the context of framed knots, of
classical knots of singular knots and of transverse knots.

Geometric Topology