Epimorphisms between 2-bridge link groups: homotopically trivial simple loops on 2-bridge spheres

Abstract

We give a complete characterization of those essential simple loops on 2-bridge spheres of 2-bridge links which are null-homotopic in the link complements. By using this result, we describe all upper-meridianpair-preserving epimorphisms between 2-bridge link groups.

Homotopically equivalent simple loops on 2-bridge spheres in Heckoid orbifolds for 2-bridge links (II)

“…The former is a contradiction to [5, Main Theorem 2.5(1)], and the latter is a contradiction to Lemma 3.7. This completes the proof of Main Theorem 1.1 (1). It remains to prove Main Theorem 1.1(3).…”

“…This implies that v 1 is cyclically alternating, becauseū s is cyclically alternating and |v 1 | = |u s | − d|u r | is even. Also since v 1 = 1 in G(K(r)), Lemma 4.1 (1) implies that the cyclic word (v 1 ) contains a proper subword w such that S(w) is (S 1 , S 2 ) or (S 2 , S 1 ). So…”

A family of two-generator non-Hopfian groups

“…In Section 2, we recall the upper presentation of a 2-bridge link group, and basic facts established in [9] concerning the upper presentations. We also recall key facts from [9] obtained by applying small cancellation theory to the upper presentations. Section 3 is devoted to the proof of the main result (Theorem 1.1).…”

Subgroups of mapping class groups related to Heegaard splittings and bridge decompositions

“…In the second author's joint work with Lee and Sakuma [18], a complete answer to the above question for 2-bridge links was given. Moreover, the following results were obtained in a series of joint work [18,20], and they were applied in [19] to give a variation of McShane's identity for 2-bridge links.…”

“…Moreover, the following results were obtained in a series of joint work [18,20], and they were applied in [19] to give a variation of McShane's identity for 2-bridge links. (See the research announcement [17] for the overview of the series of work.)…”