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

Abstract

In this paper and its two sequels, we 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. This paper treats the case when the 2-bridge link is a (2, p)-torus link, where more cases of homotopy arise, and its sequels will treat the remaining cases.

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).…”

“…By applying this identity to the identities in Proposition 6.1, we obtain the desired results. Key Lemma 6.2 is proved by using the results obtained in the series of papers [14,15,16,17] (see also the announcement [18]), which gives a complete answer to the following question concerning the simple loops in 2-bridge sphere S of a 2-bridge link K(r).…”

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].…”