New presentations of a link and virtual link

Liangxia Wan

Published 2016-04-28Version 1

New presentations of a link and a virtual link are introduced and algebraic systems on links and virtual links are constructed respectively. Based on the algebraic systems, Reduction Crossing Algorithms for them are proposed which are used to reduce the number of crossings in a link and virtual link. For known unknots, one can transform them into a trivial knot in a polynomial time by applying corresponding algorithm. As special consequences, two examples are given which are Goeritz's unknot and Thistlethwaite's unknot.