Knot theory in handlebodies

Details Knot theory in handlebodies Knot theory in handlebodies

2004-05-26

arxiv.org/abs/math/0405502v1

J. Knot Theory and Ramifications, Vol. 11, No. 6, pp. 921--943 (2002)

Mathematics

Geometric Topology

23 pages, 23 figures, LaTex file

Scientific paper

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.

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