Hyperbolic knots spanning accidental Seifert surfaces of arbitrarily high genus

Tsutsumi, Yukihiro

doi:10.1007/s00209-003-0593-0

Cited by 3 publications

(2 citation statements)

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“…Statement of the theorem. Lyon [Ly71, Theorem 2] showed that there exists a non-fibered knot K of genus one that admits incompressible Seifert surfaces of arbitrarily large genus (see also [Sce67,Gu81,Ts04] for related examples). By the discussion in Section 3, this implies that π(K) splits over free groups of arbitrarily large rank.…”

Section: Note That Inmentioning

confidence: 99%

Splittings of knot groups

2014

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Let K be a knot of genus g. If K is fibered, then it is well known that the knot group π(K) splits only over a free group of rank 2g. We show that if K is not fibered, then π(K) splits over non-free groups of arbitrarily large rank. Furthermore, if K is not fibered, then π(K) splits over every free group of rank at least 2g. However, π(K) cannot split over a group of rank less than 2g. The last statement is proved using the recent results of Agol, Przytycki-Wise and Wise.

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Section: Note That Inmentioning

confidence: 99%

Splittings of knot groups

2014

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“…Turning to surfaces with boundary, it is known that all three geometric types occur in hyperbolic link complements. For example, Tsutsumi constructed hyperbolic knots with accidental Seifert surfaces of arbitrarily high genus [25]. On the other hand, Fenley proved that minimal genus Seifert surfaces cannot be accidental [9].…”

Section: Introductionmentioning

confidence: 99%

Quasifuchsian state surfaces

Futer,

Kalfagianni,

Purcell

2012

Preprint

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This paper continues our study, initiated in [12], of essential state surfaces in link complements that satisfy a mild diagrammatic hypothesis (homogeneously adequate). For hyperbolic links, we show that the geometric type of these surfaces in the Thurston trichotomy is completely determined by a simple graph-theoretic criterion in terms of a certain spine of the surfaces. For links with A-or B-adequate diagrams, the geometric type of the surface is also completely determined by a coefficient of the colored Jones polynomial of the link.

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Quasifuchsian state surfaces

Futer

Kalfagianni

Purcell

2014

Trans. Amer. Math. Soc.

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This paper continues our study, initiated in [arXiv:1108.3370], of essential state surfaces in link complements that satisfy a mild diagrammatic hypothesis (homogeneously adequate). For hyperbolic links, we show that the geometric type of these surfaces in the Thurston trichotomy is completely determined by a simple graph--theoretic criterion in terms of a certain spine of the surfaces. For links with A- or B-adequate diagrams, the geometric type of the surface is also completely determined by a coefficient of the colored Jones polynomial of the link.Comment: 21 pages, 9 figures: To appear in the Transactions of the AM

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Hyperbolic knots spanning accidental Seifert surfaces of arbitrarily high genus

Abstract: A method for constructing hyperbolic knots each of which bounds accidental incompressible Seifert surfaces of arbitrarily high genus is given. (2000): 57N10, 57M25. Mathematics Subject Classification

Cited by 3 publications

References 20 publications

Splittings of knot groups

Splittings of knot groups

Quasifuchsian state surfaces

Quasifuchsian state surfaces

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