Universal Bounds for Genus One Seifert Surfaces for Hyperbolic Knots and Surgeries with Non-Trivial JSJT-Decompositions

Tsutsumi, Yukihiro

doi:10.4036/iis.2003.53

Cited by 11 publications

(7 citation statements)

References 6 publications

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“…Then, we have a hyperbolic knot bounding n genus 1 Seifert surfaces which are not mutually parallel. Tsutsumi first showed that the number n is at most 7 ( [41]). After that, Eudave-Muñoz -Ramírez-Losada -Valdez-Sánchez showed that n is at most 6 and provided an example of such embedding of X for n = 5 ([12]).…”

Section: Theorem 313 ([34]) For Equivalence Classesmentioning

confidence: 99%

Multibranched surfaces in 3-manifolds

Ozawa

2020

Preprint

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This is a latest survey article on embeddings of multibranched surfaces into 3-manifolds.Throughout this article, we will work in the piecewise linear category. All topological spaces are assumed to be second countable and Hausdorff. It is a fundamental problem that for two topological spaces X and Y , (1) Can X be embedded in Y ?(2) If X can be embedded in Y , then (a) When are two embeddings of X in Y equivalent? (b) How can X be embedded in Y ?In this article, we consider the case that X is a multibranched surface and Y is a closed orientable 3-manifold.We say that a 2-dimensional CW complex is a multibranched surface if we remove all points whose open neighborhoods are homeomorphic to the 2-dimensional Euclidean space, then we obtain a 1-dimensional complex which is homeomorphic to a disjoint union of some simple closed curves.

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Section: Theorem 313 ([34]) For Equivalence Classesmentioning

confidence: 99%

Multibranched surfaces in 3-manifolds

Ozawa

2020

Preprint

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“…Moreover, Kakimizu complexes are simply connected ( [189]) and contractible ( [162]). For a hyperbolic knot of genus 1, the number of mutually disjoint nonparallel genus 1 Seifert surfaces is at most 7 ( [208]). On the other hand, for any natural number n, there exists a hyperbolic knot of genus 1 which bounds mutually disjoint non-parallel Seifert surface of genus 1 and n Seifert surfaces of genus 2 ( [208]).…”

Section: Bridge Decomposing Spheresmentioning

confidence: 99%

Knots and surfaces

Ozawa

2019

Sugaku Expositions

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“…Then k is a ..1; 0/; .n; m//-curve in @H 0 (Lemma 4.3 of [Tsutsumi 2003]) with jk \ Dj D 2. See Figure 27.…”

Section: Figure 26mentioning

confidence: 99%

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2015

Pacific J. Math.

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In this paper we give a combinatorial characterization of tight fusion frame (TFF) sequences using Littlewood-Richardson skew tableaux. The equal rank case has been solved recently by Casazza, Fickus, Mixon, Wang, and Zhou. Our characterization does not have this limitation. We also develop some methods for generating TFF sequences. The basic technique is a majorization principle for TFF sequences combined with spatial and Naimark dualities. We use these methods and our characterization to give necessary and sufficient conditions which are satisfied by the first three highest ranks. We also give a combinatorial interpretation of spatial and Naimark dualities in terms of Littlewood-Richardson coefficients. We exhibit four classes of TFF sequences which have unique maximal elements with respect to majorization partial order. Finally, we give several examples illustrating our techniques including an example of tight fusion frame which can not be constructed by the existing spectral tetris techniques. We end the paper by giving a complete list of maximal TFF sequences in dimensions ≤ 9.

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Universal Bounds for Genus One Seifert Surfaces for Hyperbolic Knots and Surgeries with Non-Trivial JSJT-Decompositions

Abstract: A universal bound on the number of mutually disjoint non-parallel genus one Seifert surfaces for hyperbolic knots in non-Haken manifolds is given.

Cited by 11 publications

References 6 publications

Multibranched surfaces in 3-manifolds

Multibranched surfaces in 3-manifolds

Knots and surfaces

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