Twisted conjugacy in dihedral Artin groups I: Torus Knot groups
Gemma Crowe
TL;DR
This work proves that the twisted conjugacy problem for odd dihedral Artin groups $G(m)$ is solvable in linear time by exploiting an isomorphism to the torus knot group $P=ig\<x,yig angle$ with $x^{2}=y^{m}$ and the finite outer automorphism group $ ext{Out}(G(m))cong C_{2}$. Central to the method are geodesic normal forms (via Fujii) and a Garside-based framework that produces Garside-free, minimal twisted-conjugacy representatives; these representations are connected by a finite sequence of $ heta$-cyclic permutations and equivalences, enabling efficient decision procedures. The authors also show that the $k$-simultaneous conjugacy problem is solvable in $G(m)$ and use orbit decidability results to deduce that extensions $G(m) times H$ with suitable $H$ have decidable conjugacy problems, broadening the class of groups with practical CP/TCP algorithms. Together, these results advance the algorithmic theory of Artin groups, provide implementable procedures, and yield concrete consequences for the conjugacy problem in group extensions. The work highlights how a torus knot perspective and Garside-like techniques can yield sharp, scalable decision procedures in a historically challenging class of groups.
Abstract
In this paper we provide an alternative solution to a result by Juhász that the twisted conjugacy problem for odd dihedral Artin groups is solvable, that is, groups with presentation $G(m) = \langle a,b \; | \; _{m}(a,b) = {}_{m}(b,a) \rangle$, where $m\geq 3$ is odd, and $_{m}(a,b)$ is the word $abab \dots$ of length $m$, is solvable. Our solution provides an implementable linear time algorithm, by considering an alternative group presentation to that of a torus knot group, and working with geodesic normal forms. An application of this result is that the conjugacy problem is solvable in extensions of odd dihedral Artin groups.