Representations of branched twist spins with a non-trivial center of order 2

2022/09/23 by Mizuki Fukuda, Fukuda, Mizuki
Computer Science · Mathematics · #57M27(Secondary) #57Q45 (Primary) 57M60 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2209.11583

openalex publication_date 2022/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is known that a presentation of the knot group of a branched twist spin is obtained from a Wirtinger presentation of the original 1-knot group by adding a generator corresponding to a regular orbit of the circle action and a certain relator. In particular, the additional generator is an element of the center of the knot group. In this paper, we focus on \rm SL₂(ℤ₃)-representations and dihedral group representations. For the former case, we give a sufficient condition for the existence of an \rm SL₂(ℤ₃)-representation for a branched twist spin. For the latter case, we determine the number of even-ordered dihedral group representations of branched twist spins.

Related

• Irreducible SL(2,ℂ)-metabelian representations of branched twist spins
• Distinguishing 2-knots admitting circle actions by fundamental groups
• The groups of branched twist-spun knots
• Branched twist spins and knot determinants
• Dihedral twists in the Twist Conjecture