Guts of surfaces in punctured-torus bundles

Kuessner, Thilo

doi:10.1007/s00013-005-1097-4

Cited by 8 publications

(5 citation statements)

References 12 publications

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“…To date, there have only been a handful of results computing the guts of essential surfaces in an infinite family of manifolds: see e.g. [3,57,58]. In particular, Lackenby's computation of the guts of checkerboard surfaces of alternating links [58,Theorem 5] In the A-adequate setting, we have the following volume estimate.…”

Section: Volume Bounds For Hyperbolic Linksmentioning

confidence: 99%

“…For instance, Lackenby's diagrammatic lower bound on the volumes of alternating knots and links came as a result of computing the guts of checkerboard surfaces [58]. However, computing χ − (guts) has typically been quite hard: apart from alternating knots and links, there are very few infinite families of manifolds for which there are known computations of the guts of essential surface [3,57].…”

Section: Volume Bounds From Topology and Combinatoricsmentioning

confidence: 99%

“…See [49]. Prior to this manuscript, there have been only a few cases in which the guts of essential surfaces have been explicitly understood and calculated for an infinite family of 3-manifolds [3,57,58]. Affirmative answers to Questions 10.2, 10.3, and 10.4 would add to the list of these results, and could have further applications.…”

Section: Hyperbolic Volume and Colored Jones Polynomialsmentioning

confidence: 99%

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Guts of Surfaces and the Colored Jones Polynomial

Futer

Kalfagianni

Purcell

2013

Lecture Notes in Mathematics

198

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PrefaceAround 1980, W. Thurston proved that every knot complement satisfies the geometrization conjecture: it decomposes into pieces that admit locally homogeneous geometric structures. In addition, he proved that the complement of any non-torus, non-satellite knot admits a complete hyperbolic metric which, by the Mostow-Prasad rigidity theorem, is is necessarily unique up to isometry. As a result, geometric information about a knot complement, such as its volume, gives topological invariants of the knot.Since the mid-1980's, knot theory has also been invigorated by ideas from quantum physics, which have led to powerful and subtle knot invariants, including the Jones polynomial and its relatives, the colored Jones polynomials. Topological quantum field theory predicts that these quantum invariants are very closely connected to geometric structures on knot complements, and particularly to hyperbolic geometry. The volume conjecture of R. Kashaev, H. Murakami, and J. Murakami, which asserts that the volume of a hyperbolic knot is determined by certain asymptotics of colored Jones polynomials, fits into the context of these predictions. Despite compelling experimental evidence, these conjectures are currently verified for only a few examples of hyperbolic knots.This monograph initiates a systematic study of relations between quantum and geometric knot invariants. Under mild diagrammatic hypotheses that arise naturally in the study of knot polynomial invariants (A-or B-adequacy), we derive direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. We prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement, and that certain coefficients of the polynomial measure how far this surface is from being a fiber in the knot complement. In particular, the surface is a fiber if and only if a certain coefficient vanishes.Our results also yield concrete relations between hyperbolic geometry and colored Jones polynomials: for certain families of links, coefficients of the polynomials determine the hyperbolic volume to within a factor of 4. Our methods here provide a deeper and more intrinsic explanation for similar connections that have been previously observed.Our approach is to generalize the checkerboard decompositions of alternating knots and links. For A-or B-adequate diagrams, we show that the checkerboard knot surfaces are incompressible, and obtain an ideal polyhedral decomposition of their complement. We use normal surface theory to establish a dictionary between the pieces of the JSJ decomposition of the surface complement and the combinatorial structure of certain spines of the checkerboard surface (state graphs). In particular, we give a combinatorial formula 4 for the complexity of the hyperbolic part of the JSJ decomposition (the guts) of the surface complement in terms of the diagram of the knot, and use this to give lower bounds on volumes of several ...

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Section: Volume Bounds For Hyperbolic Linksmentioning

confidence: 99%

Section: Volume Bounds From Topology and Combinatoricsmentioning

confidence: 99%

Section: Hyperbolic Volume and Colored Jones Polynomialsmentioning

confidence: 99%

See 1 more Smart Citation

Guts of Surfaces and the Colored Jones Polynomial

Futer

Kalfagianni

Purcell

2013

Lecture Notes in Mathematics

198

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“…In [13,Corollary 5.19] we proved that, when all 2-edge loops in G A belong to twist regions, −χ(guts(S 3 K, S A )) = max{−χ(G ′ A ), 0}. Furthermore, the work of Agol [4], as generalized by Kuessner [21], says that guts can be used to estimate the Gromov norm of S 3 K:…”

Section: Proof Let G ′mentioning

confidence: 99%

Hyperbolic semi-adequate links

Futer

Kalfagianni

Purcell

2015

Communications in Analysis and Geometry

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“…Recall that the guts of M \\S is defined to be the complement of the maximal Ibundle in M \\S. Work of Agol [1], extended by Kuessner [10], says that guts can be used to bound the Gromov norm of M . In particular, (1) ||M || ≥ 2 χ − (guts(M\\S)).…”

Section: Volume Applicationsmentioning

confidence: 99%