On the Fibonacci Groups, the Tbrk's Head Links and Hyperbolic 3-Manifolds

Mednykh, Alexander; Веснин, Андрей Юрьевич

doi:10.1515/9783110908978.231

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“…Kim [VesK98a] consider 2-fold branched covers over certain 3-braids. Alexander Mednykh and Andrei Vesnin [MedV95] consider 2-fold branched covers over Turk's head links.…”

Section: Historymentioning

confidence: 99%

Bitwist manifolds and two-bridge knots

et al. 2016

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We give uniform, explicit, and simple face-pairing descriptions of all the branched cyclic covers of the 3-sphere, branched over two-bridge knots. Our method is to use the bi-twisted face-pairing constructions of Cannon, Floyd, and Parry; these examples show that the bi-twist construction is often efficient and natural. Finally, we give applications to computations of fundamental groups and homology of these branched cyclic covers. arXiv:1306.4564v2 [math.GT] 10 Apr 2015Remark 1.1. The 2's in the continued fraction indicate that the tangle is constructed using only full twists instead of the possible mixture of full and half twists. Example 1.2. The simplest case, with only equator and longitude, yields the trefoil and figureeight knots, as we shall see in Theorem 4.1. Simple subdivisions yield their branched cyclic covers, the Sieradski [Sie86] and Fibonacci [VesM96a] manifolds.Definition 1.3. We say that the multiplier function m is normalized if the following hold:(1) m 2k+1 = 0, and (2) if m 2i+1 = 0 for some i ∈ {0, . . . , k − 1}, then m 2i = m 2i+2 .With this definition, the previous theorem and well-known results involving two-bridge knots yield the following corollary.Corollary 4.4. Every normalized multiplier function yields a nontrivial two-bridge knot. Conversely, every nontrivial two-bridge knot K is realized by either one or two normalized multiplier functions. If K is the numerator closure of the tangle T (a/b), then it has exactly one such realization if and only if b 2 ≡ 1 mod a.Notice that the n-th branched cyclic covering of S 3 , branched over K, can be obtained by unwinding the description n times about the unknotted axis that represents K, unwinding the initial face-pairing as in Figure 17. This has the following application.Theorem 5.2. The fundamental group of the n-th branched cyclic covering of S 3 , branched over K, has a cyclic presentation.Problem 1.4. How should one carry out the analogous construction for arbitrary knots?

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“…Kim [VesK98a] consider 2-fold branched covers over certain 3-braids. Alexander Mednykh and Andrei Vesnin [MedV95] consider 2-fold branched covers over Turk's head links.…”

Section: Historymentioning

confidence: 99%

Bitwist manifolds and two-bridge knots

et al. 2016

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“…Maclachlan and A. Reid [1997] and Vesnin and A. C. considered 2-fold branched covers over certain 3-braids. Alexander Mednykh and Vesnin [1995] considered 2-fold branched covers over Turk's head links.…”

Section: Ementioning

confidence: 99%

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2016

Pacific J. Math.

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Though LeeR Lambert spent his life as an actuary and a musician and was a loving father of nine girls and one boy, he had always wanted to earn an advanced degree as a mathematician. With the encouragement of his wife, he earned his Ph.D. in mathematics at the age of 68. Many of the results of this paper appeared in his Ph.D. dissertation at Brigham Young University. At the age of 71, LeeR died of bone cancer. We miss you, LeeR. We give uniform, explicit, and simple face-pairing descriptions of all the branched cyclic covers of the 3-sphere, branched over two-bridge knots. Our method is to use the bitwisted face-pairing constructions of Cannon, Floyd, and Parry; these examples show that the bitwist construction is often efficient and natural. Finally, we give applications to computations of fundamental groups and homology of these branched cyclic covers.

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On the Fibonacci Groups, the Tbrk's Head Links and Hyperbolic 3-Manifolds

Cited by 2 publications

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Bitwist manifolds and two-bridge knots

Bitwist manifolds and two-bridge knots

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