A Survey of Hyperbolic Knot Theory

Futer, David; Kalfagianni, Efstratia; Purcell, Jessica S.

doi:10.1007/978-3-030-16031-9_1

Cited by 4 publications

(5 citation statements)

References 76 publications

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“…the interior of the 3manifold C(K 9 46 ) admits a hyperbolic metric. In general, every link L is hyperbolic if the interior of its complement C(L) is a hyperbolic 3-manifold (with negative sectional curvature), see [21] for a recent survey. There is a deep result of Menasco [22], that every link having a knot diagram with alternating crossings along each components is a hyperbolic link if it is nonsplit (every component is linked) and prime (it cannot be decomposed into sums) unless it is a torus link.…”

Section: Geometric Properties Of the Embeddingmentioning

confidence: 99%

How to obtain a cosmological constant from small exotic R4

Asselmeyer-Maluga

Król

2018

Physics of the Dark Universe

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In this paper we determine the cosmological constant as a topological invariant by applying certain techniques from low dimensional differential topology. We work with a small exotic R 4 which is embedded into the standard R 4 . Any exotic R 4 is a Riemannian smooth manifold with necessary non-vanishing curvature tensor. To determine the invariant part of such curvature we deal with a canonical construction of R 4 where it appears as a part of the complex surface K3#CP (2). Such R 4 's admit hyperbolic geometry. This fact simplifies significantly the calculations and enforces the rigidity of the expressions. In particular, we explain the smallness of the cosmological constant with a value consisting of a combination of (natural) topological invariant. Finally, the cosmological constant appears to be a topologically supported quantity.

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Section: Geometric Properties Of the Embeddingmentioning

confidence: 99%

How to obtain a cosmological constant from small exotic R4

Asselmeyer-Maluga

Król

2018

Physics of the Dark Universe

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“…These two topological polyhedra do not necessarily agree with the complete hyperbolic structure. That is, there may not exist two ideal hyperbolic polyhedra having the same combinatorial type as checkerboard polyhedra and they can be glued as checkerboard polyhedra to give the complete hyperbolic structure of S 3 − L. We define horoball neighborhood of a link following [13].…”

Section: Definitionsmentioning

confidence: 99%

Alternating links with totally geodesic checkerboard surfaces

Gan

2020

Preprint

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“…To state our results we need some terminology that we will not define in detail. For definitions and more details the reader is referred to [15]. Over the years there has been a number of results about coarse relations between diagrammatic link invariants and the volume of hyperbolic links.…”

Section: Bounds For Seifert Manifoldsmentioning

confidence: 99%

“…Over the years there has been a number of results about coarse relations between diagrammatic link invariants and the volume of hyperbolic links. See [15] and references therein. Using such results, for restricted classes of 3-manifolds, we obtain sharper bounds than the one of Theorem 1.1.…”

Section: Bounds For Seifert Manifoldsmentioning

confidence: 99%

“…The problem of estimating the volume of hyperbolic 3-manifolds in terms of topological quantities and quantum invariants, has been studied considerably in the literature. See for example [1,14,15] and references in the last item. Despite progress, to the best of our knowledge, Theorem 1.1 gives the first such linear lower bound that works for all hyperbolic 3-manifolds.…”

Section: Introductionmentioning

confidence: 99%

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Gromov norm and Turaev-Viro invariants of 3-manifolds

Detcherry,

Kalfagianni

2017

Preprint

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We establish a relation between the "large r" asymptotics of the Turaev-Viro invariants T Vr and the Gromov norm of 3-manifolds. We show that for any orientable, compact 3-manifold M , with (possibly empty) toroidal boundary, log |T Vr(M )| is bounded above by a function linear in r and whose slope is a positive universal constant times the Gromov norm of M . The proof combines TQFT techniques, geometric decomposition theory of 3-manifolds and analytical estimates of 6j-symbols.We obtain topological criteria that can be used to check whether the growth is actually exponential; that is one has log |T Vr(M )| B r, for some B > 0. We use these criteria to construct infinite families of hyperbolic 3-manifolds whose SO(3) Turaev-Viro invariants grow exponentially. These constructions are essential for the results of [10] where the authors make progress on a conjecture of Andersen, Masbaum and Ueno about the geometric properties of surface mapping class groups detected by the quantum representations.We also study the behavior of the Turaev-Viro invariants under cutting and gluing of 3-manifolds along tori. In particular, we show that, like the Gromov norm, the values of the invariants do not increase under Dehn filling and we give applications of this result on the question of the extent to which relations between the invariants T Vr and hyperbolic volume are preserved under Dehn filling.Finally we give constructions of 3-manifolds, both with zero and non-zero Gromov norm, for which the Turaev-Viro invariants determine the Gromov norm.

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A Survey of Hyperbolic Knot Theory

Cited by 4 publications

References 76 publications

How to obtain a cosmological constant from small exotic R4

How to obtain a cosmological constant from small exotic R4

Alternating links with totally geodesic checkerboard surfaces

Gromov norm and Turaev-Viro invariants of 3-manifolds

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