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

2013

Geom Dedicata

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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

Section: Resultsmentioning

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Section: Resultsmentioning

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Section: Introductionmentioning

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See 3 more Smart Citations

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

2013

Geom Dedicata

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

2013

Geom Dedicata

Self Cite

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.

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

Proceedings of the London Mathematical Society

2011

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107