  In this paper we study properties of the Markov trace ${\rm tr}_d$ and the
specialized trace ${\rm tr}_{d,D}$ on the Yokonuma-Hecke algebras, such as
behaviour under inversion of a word, connected sums and mirror imaging. We then
define invariants for framed, classical and singular links through the trace
${\rm tr}_{d,D}$ and also invariants for transverse links through the trace
${\rm tr}_d$. In order to compare the invariants for classical links with the
Homflypt polynomial we develop computer programs and we evaluate them on
several Homflypt-equivalent pairs of knots and links. Our computations lead to
the result that these invariants are topologically equivalent to the Homflypt
polynomial on knots. However, they do not demonstrate the same behaviour on
links.
