The mixed Hilden braid group and the plat equivalence in handlebodies

2023/12/17 by Cavicchioli, Paolo, Lambropoulou, Sofia
#57K10 #57K12 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2312.10781

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.

Related

• Knot theory in handlebodies
• Passing from plat closure to standard closure of braids in ℝ³, in handlebodies and in thickened surfaces
• The Hilden Double Coset Problem in Braid Groups
• Pseudo links in handlebodies
• Hilden Braid Groups