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

Abstract

This is the second 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. In this paper, we treat the case of 2-bridge links of slope n/(2n + 1) and (n + 1)/(3n + 2), where n ≥ 2 is an arbitrary integer.knot, is the most complicated. In fact, the figure-eight knot group admits various unexpected reduced annular diagrams (see Section 7). This reminds us of the phenomenon in the theory of exceptional Dehn filling that the figure-eight knot attains the maximal number of exceptional Dehn fillings.This paper is organized as follows. In Section 2, we describe the main results of this paper (Main Theorems 2.2 and 2.3). In Section 3, we set up Hypotheses A, B and C, under which we establish technical lemmas used for the proofs in Sections 4-6. The special case of Main Theorem 2.2 (namely, the case of a 2-bridge link of slope 2/5) is treated in Section 4, and the remaining case of Main Theorem 2.2 (namely, the case of a 2-bridge link of slope n/(2n+1) with n ≥ 3) in Section 5. The proof of Main Theorem 2.3 is contained in Section 6. In the final section, Section 7, we prove Theorems 2.5 and 2.6. Main resultsThis paper, as a continuation of [2], uses the same notation and terminology as in [2] without specifically mentioning. We begin with the following question, providing whose answer is the purpose of this series of papers.Question 2.1. Consider a 2-bridge link K(r) with r = ∞. For two distinct rational numbers s, s ′ ∈ I 1 (r) ∪ I 2 (r), when are the unoriented loops α s and α s ′ homotopic in S 3 − K(r)?In the first paper [2], we treated the case when r = 1/p for some p ∈ Z, and obtained a complete answer (see [2, Main Theorem 2.7]). In the present paper, we solve the above question for the 2-bridge links K(n/(2n + 1)) and K((n + 1)/(3n + 2)), where n ≥ 2 is an arbitrary integer.Main Theorem 2.2. Suppose r = n/(2n + 1) = [2, n], where n ≥ 2 is an integer. Then, for any two distinct rational numbers s, s ′ ∈ I 1 (r) ∪ I 2 (r), the unoriented loops α s and α s ′ are never homotopic in S 3 − K(r).Main Theorem 2.3. Suppose r = (n + 1)/(3n + 2) = [2, 1, n], where n ≥ 2 is an integer. Then, for two distinct rational numbers s, s ′ ∈ I 1 (r) ∪ I 2 (r), the unoriented loops α s and α s ′ are homotopic in S 3 − K(r) if and only if both r = 3/8 (i.e., n = 2) and the set {s, s ′ } equals either {1/6, 3/10} or {3/4, 5/12}. Remark 2.4. The exceptional pairs {1/6, 3/10} and {3/4, 5/12} have the following geometric properties in the Farey tessellation. Let τ be the reflection of the hyperbolic plane in the geodesic with endpoints 1/2 and 1/4, which bisects the Farey edge 0/1, 1/3 . Then τ preserves the Farey tessellation and 2 4 25

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).…”

“…(He denotes the set by the symbol L(φ), where φ is a Markoff map.) By using the classification of the essential simple loops in the 2-bridge sphere which are peripheral in hyperbolic 2-bridge links complements (see [16,17] and [18, Theorem 2.6(1)]), we have the following theorem for Bowdich's set of end invariants L(ρ r ) := L(φ 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].…”