Twisted conjugacy in dihedral Artin groups I: Torus Knot groups

In this paper we provide an alternative solution to a result by Juh\'asz that the twisted conjugacy problem for odd dihedral Artin groups is solvable, that is, groups with presentation G (m) = a, b \; | \; ₌ (a, b) = ₌ (b, a), where m 3 is odd, and ₌ (a, b) is the word abab 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.