  It is a natural question to ask whether two links are equivalent by the
following moves -- parallel parts of a link are changed to k-times half-twisted
parts and if they are, how many moves are needed to go from one link to the
other. In particular if k=2 and the second link is a trivial link it is the
question about the unknotting number. The new polynomial invariants of links
often allow us to answer the above questions. Also the first homology groups of
cyclic branch covers over links provide some interesting information. In the
first part of the paper we apply the Jones-Conway (Homflypt) and Kauffman
polynomials to find whether two links are not t_k equivalent and if they are,
to gain some information how many moves are needed to go from one link to the
other. In the second part we describe the Fox congruence classes and their
relations with t_k moves. We use the Fox method to analyze relations between
t_k moves and the first homology groups of branched cyclic covers of links.In
the third part we consider the influence of t_k moves on the Goeritz and
Seifert matrices and analyze Lickorish-Millett and Murakami formulas from the
point of view of t_k moves and illustrate them by various examples. At the end
of the paper we outline some relations with signatures of links and non-cyclic
coverings of link spaces.
