Twisted conjugacy in dihedral Artin groups I: Torus Knot groups

In this paper we provide an alternative solution to a result by (Juh´asz 2011), that the twisted conjugacy problem for odd dihedral Artin groups is solvable, where an odd dihedral Artin group is a group with presentation G(m) = ⟨a, b | _m(a, b) = _m(b, a)⟩, where m ≥ 3 is odd, and _m(a, b) is the word abab . . . of length m. 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.