Orderability and Dehn filling

Culler, Marc; Dunfield, Nathan M.

doi:10.2140/gt.2018.22.1405

Cited by 66 publications

(95 citation statements)

References 51 publications

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“…We conclude this section with further numerical experiments, concerning the distribution of the hyperbolic volume in the Petaluma model, as approximated by the Sage package SnapPy (Culler et al 2016).…”

Section: Hyperbolic Volumementioning

confidence: 97%

“…Other invariants such as the Alexander-Conway polynomial and finite type invariants are computable in polynomial time (Alexander 1928;Bar-Natan 1995a;Chmutov et al 2012). Many such algorithms are implemented in software packages, such as SnapPy (Culler et al 2016), KnotTheory (Bar-Natan et al 2016b) and KnotScape (Hoste and Thistlethwaite 2016). These are used in practice for the compilation of knot databases and are important tools in research and applications (Bar-Natan et al 2016a;Cha and Livingston 2016;Jim Hoste et al 1998).…”

Section: Computational Aspectsmentioning

confidence: 99%

“…Their results suggest that several invariants, including the hyperbolic volume, grow linearly with n. and Chapman (2016c) precisely compute knot probabilities for n 10, and study their behavior for larger n based on random samples. The methods used in these experiments are implemented into publicly available software packages: plCurve (Ashton et al 2016) and SnapPy (Culler et al 2016). …”

Section: This Extends Pattern Theorems Formentioning

confidence: 99%

“…14 was obtained from the Sage software SnapPy (Culler et al 2016), that approximates the hyperbolic volume of a link by finding a triangulation of its complement with compatible hyperbolic structure. In order to make the random samples suitable as input for the program, we first converted them from petal diagrams to braids, as shown in Fig.…”

Section: Implementation Detailsmentioning

confidence: 99%

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Models of random knots

Even-Zohar

2017

J Appl. and Comput. Topology

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The study of knots and links from a probabilistic viewpoint provides insight into the behavior of ''typical'' knots, and opens avenues for new constructions of knots and other topological objects with interesting properties. The knotting of random curves arises also in applications to the natural sciences, such as in the context of the structure of polymers. We present here several known and new randomized models of knots and links. We review the main known results on the knot distribution in each model. We discuss the nature of these models and the properties of the knots they produce. Of particular interest to us are finite type invariants of random knots, and the recently studied Petaluma model. We report on rigorous results and numerical experiments concerning the asymptotic distribution of such knot invariants. Our approach raises questions of universality and classification of the various random knot models.

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Section: Hyperbolic Volumementioning

confidence: 97%

Section: Computational Aspectsmentioning

confidence: 99%

Section: This Extends Pattern Theorems Formentioning

confidence: 99%

Section: Implementation Detailsmentioning

confidence: 99%

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Models of random knots

Even-Zohar

2017

J Appl. and Comput. Topology

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“…In SnapPy's census [39], the 3-manifold is denoted as m003 and allows an ideal triangulation using two tetrahedrons (see appendix A):…”

Section: Jhep08(2017)118mentioning

confidence: 99%

3d N = 2 $$ \mathcal{N}=2 $$ minimal SCFTs from wrapped M5-branes

Bae

Gang

Lee

2017

J. High Energ. Phys.

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Abstract:We study CFT data of 3-dimensional superconformal field theories (SCFTs) arising from wrapped two M5-branes on closed hyperbolic 3-manifolds. Via so-called 3d/3d correspondence, central charges of these SCFTs are related to a SL(2) Chern-Simons (CS) invariant on the 3-manifolds. After developing a state-integral model for the invariant, we numerically evaluate the central charges for several closed 3-manifolds with small hyperbolic volume. The computation suggests that the wrapped M5-brane systems give infinitely many discrete SCFTs with small central charges.

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An Upper Bound on Pachner Moves Relating Geometric Triangulations

Kalelkar

Phanse

2021

Discrete Comput Geom

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Suppose that M is a compact, connected three-manifold with boundary. We show that if the universal cover has infinitely many boundary components then M has an ideal triangulation which is essential: no edge can be homotoped into the boundary. Under the same hypotheses, we show that the set of essential triangulations of M is connected via 2-3, 3-2, 0-2, and 2-0 moves.The above results are special cases of our general theory. We introduce L-essential triangulations: boundary components of the universal cover receive labels and no edge has the same label at both ends. As an application, under mild conditions on a representation, we construct an ideal triangulation for which a solution to Thurston's gluing equations recovers the given representation.Our results also imply that such triangulations are connected via 2-3, 3-2, 0-2, and 2-0 moves. Together with results of Pandey and Wong, this proves that Dimofte and Garoufalidis' 1-loop invariant is independent of the choice of essential triangulation.

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Orderability and Dehn filling

Cited by 66 publications

References 51 publications

Models of random knots

Models of random knots

3d N = 2 $$ \mathcal{N}=2 $$ minimal SCFTs from wrapped M5-branes

An Upper Bound on Pachner Moves Relating Geometric Triangulations

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