t_kmoves on links
Abstract
It is a natural question to ask whether two links are equivalent by the following moves  parallel parts of a link are changed to ktimes halftwisted 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 JonesConway (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 LickorishMillett 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 noncyclic coverings of link spaces.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606633
 Bibcode:
 2006math......6633P
 Keywords:

 Mathematics  Geometric Topology;
 57M27 (primary)
 EPrint:
 50 pages, 44 figures