Knot theory in handlebodies

2004/05/26 by Reinhard Haering-Oldenburg, Sofia Lambropoulou
Mathematics · #math.GT #math.AT #msc:57M25

paper · pdf

published as J. Knot Theory and Ramifications, Vol. 11, No. 6, pp. 921--943 (2002) · 23 pages, 23 figures, LaTex file

arxiv created 2004/05/26 · arxiv updated 2009/12/01

Abstract

We consider oriented knots and links in a handlebody of genus g through appropriate braid representatives in S³, 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.

Related

• Algebraic Markov equivalence for links in 3-manifolds
• The mixed Hilden braid group and the plat equivalence in handlebodies
• On Markov's Theorem
• Alexander and Markov theorems for generalized knots, II generalized braids
• Braid equivalences and the L--moves