Two-Bridge Knots with Property Q

Simon’s conjecture for two-bridge knots

“…There is a homomorphism ψ : π 1 (S 3 \ K ) → C r,s (where as in Section 3.1, C r,s denotes the free product of two cyclic groups of orders r and s) and generators a of Z/r and b of Z/s for which ψ(μ ) = ab. Theorem 2.1 of [10] and the remark following it, shows that one of r or s equals 2, say r = 2. In particular K is a two-bridge torus knot.…”

“…Indeed, using [2,3,10,12], one can say more about the nature of the homomorphisms in Theorem 1.2. In particular, we establish: Corollary 1.3.…”

“…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 .…”

Normal forms of generating pairs for Fuchsian and related groups

“…As an application we prove the following result (Theorem 18) used in [9], which originally motivated our studies. Let Π be a group with generators a and b with a conjugate to b.…”

“…Define m, n, p and q as in Lemma 11 and let Remark: Using Lemma 10 the algorithm can be improved by deciding, for most of the cases, which of the matrices M, 1 0 0 −1 M should be used as N . We close this section with an application used in [9]. Remark: The proposition states, in particular, that Z a * Z b cannot be generated by two conjugate elements if a > 2 and b > 2.…”