Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (III)

Abstract

This is the last of a series of papers which give a necessary and sufficient condition for two essential simple loops on a 2-bridge sphere in a 2-bridge link complement to be homotopic in the link complement. The first paper of the series treated the case of the 2-bridge torus links, and the second paper treated the case of 2-bridge links of slope n/(2n + 1) and (n + 1)/(3n+2), where n ≥ 2 is an arbitrary integer. In this paper, we first treat the case of 2-bridge links of slope n/(mn + 1) and (n + 1)/((m + 1)n + m), where m ≥ 3 is an arbitrary integer, and then treat the remaining cases by induction.

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

“…This is an analogy of a natural question for 2-bridge links, which has the origin in Minsky's question [3,Question 5.4], and which was completely solved in the series of papers [6,11,12,13] and applied in [8]. See [5] for an overview of these works and [16] for a related work.…”

A variation of McShane’s identity for 2–bridge links

“…Moreover, it is proved by [15,16,17] that generically two simple loops α s and α s ′ with s, s ′ ∈ I 1 (r) ∪ I 2 (r) are homotopic in the link complement only when s = s ′ (see Theorem 6.3). Now assume r = q/p, where p and q are relatively prime positive integers such that q ≡ ±1 (mod p).…”

Epimorphisms from 2-bridge link groups onto Heckoid groups (II)

“…This paper and its sequel [9] are concerned with the following natural question, which is an analogy of [2, Question 1.1] that is completely solved in the series of papers [3,4,5,6] and applied in [7].…”