The mixed Hilden braid group and the plat equivalence in handlebodies

Abstract

Given a knot or link in the handlebody, $H_g$, of genus $g$ we prove that it can always be represented as the plat closure of a braid in $H_g$. We further establish the Hilden braid group for the handlebody, as a subgroup of the mixed braid group, whose elements have the first $g$ strands fixed forming the identity braid. We then formulate and prove the algebraic equivalence connecting mixed plats belonging to the same link isotopy class in $H_g$. The plat closure representation can be particularly suitable for computing knot invariants.