Knot theory in handlebodies

Reinhard Haering-Oldenburg; Sofia Lambropoulou

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Knot theory in handlebodies

Reinhard Haering-Oldenburg·Sofia Lambropoulou

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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.

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BibTeX RIS

Reinhard Haering-Oldenburg, Sofia Lambropoulou. 2004-05-26. Knot theory in handlebodies. https://arxiv.org/abs/math/0405502

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