Normal forms of generating pairs for Fuchsian and related groups

Simon’s conjecture for two-bridge knots

“…Moreover, this 2-orbifold group is generated by the images of the two conjugate meridians of K. It is easily seen that this forces the base orbifold to be a disc with two cone points. It now follows from [14,Proposition 17] that the base orbifold group is C 2,s where s is odd, and the proof is completed as before. ⊔ ⊓ Remark: Note that the paper [24] gives a systematic construction of epimorphisms between 2-bridge knot groups.…”

“…To that end, Theorem 1.2 of [10] shows that there is an isomorphism θ : C 2,s → C 2,s such that θψϕ(μ) = ab m for some integer m. Up to inner isomorphism, we can suppose that θ(a) = a and θ(b) = b k for some k coprime with s (see, for example, [11,Theorem 13(1), Corollary 14]). Thus we can assume that ψϕ(μ) = ab m .…”

Conjugate Generators of Knot and Link Groups

“…We begin our investigation of conjugate generators for torus knot groups by paraphrasing Proposition 17 of [8]: Proposition 5.1. If Z p * Z q = s, t | s p = t q = 1 can be generated by two conjugate elements, then p = 2.…”

“…By [8,Proposition 17], the (p, q)-torus knot group can be generated by two conjugate elements only when p = 2, i.e. when the torus knot is two-bridge.…”

“…A knot in S 3 is a hyperbolic, satellite, or torus knot (Corollary 2.5 of [18]). By Proposition 17 of [8], the (p, q)-torus knot group can be generated by two conjugate elements only when p = 2, i.e., when the torus knot is two-bridge. In Section 5, we establish Reid's Conjecture for the (2, 3)-torus knot group (i.e., the trefoil knot group), and we prove in Section 6 that Reid's Conjecture implies Simon's Conjecture for two-bridge knots.…”