KNOT THEORY IN HANDLEBODIES

Abstract

We consider oriented knots and links in a handlebody of genus g through appropriate braid representatives in S 3 , which are elements of the braid groups B g,n . We prove a geometric version of the Markov theorem for braid equivalence in the handlebody, which is based on the L-moves. Using this we then prove two algebraic versions of the Markov theorem. The first one uses the L-moves. The second one uses the Markov moves and conjugation in the groups B g,n . We show that not all conjugations correspond to isotopies.