Twisted conjugacy in dihedral Artin groups I: Torus Knot groups

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 $G(m)=⟨a,b\mathrm{\mid }{}_m(a,b)={}_m(b,a)⟩$, where $m\ge 3$ 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.