Combinatorial Cubings, Cusps, and the Dodecahedral Knots

Aitchison, Iain R.; Rubinstein, J. Hyam

doi:10.1515/9783110857726.17

Cited by 30 publications

(44 citation statements)

References 4 publications

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“…Hidden symmetries A pair of commensurable knot complements that do not arise as above are the two dodecahedral knot complements of Aitchison and Rubinstein [1]. These are commensurable of the same volume, and have hidden symmetries (see Neumann and Reid [17]).…”

Section: Commensurability Classes Containing More Than One Knot Complmentioning

confidence: 99%

Commensurability classes of 2–bridge knot complements

Reid

Walsh

2008

Algebr. Geom. Topol.

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Section: Commensurability Classes Containing More Than One Knot Complmentioning

confidence: 99%

Commensurability classes of 2–bridge knot complements

Reid

Walsh

2008

Algebr. Geom. Topol.

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“…This has one torus cusp with shape parameter z = 6i generating a cusp field Q(i) which is strictly contained in its invariant trace field Q(i, √ 3). This answers a question of Neumann-Reid in [27], who asked whether the figure eight knot and the two dodecahedral knots of Aitchison-Rubinstein [3] were the only such examples 2 .…”

Section: Commensurability Of Cuspsmentioning

confidence: 70%

Commensurators of Cusped Hyperbolic Manifolds

Goodman

Heard

Hodgson

2008

Experimental Mathematics

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Abstract. This paper describes a general algorithm for finding the commensurator of a non-arithmetic hyperbolic manifold with cusps, and for deciding when two such manifolds are commensurable. The method is based on some elementary observations regarding horosphere packings and canonical cell decompositions. For example, we use this to find the commensurators of all non-arithmetic hyperbolic once-punctured torus bundles over the circle.For hyperbolic 3-manifolds, the algorithm has been implemented using Goodman's computer program Snap. We use this to determine the commensurability classes of all cusped hyperbolic 3-manifolds triangulated using at most 7 ideal tetrahedra, and for the complements of hyperbolic knots and links with up to 12 crossings.

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“…We now turn our attention to the dodecahedral knots D f and Ds as described by Aitchison and Rubinstein [1]. These two knots exhibit remarkable properties [2,26], and each can be represented with 20 crossings [2].…”

Section: The Dodecahedral Knotsmentioning

confidence: 98%

Computing closed essential surfaces in knot complements

Burton

Coward

Tillmann

2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

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We present a new, practical algorithm to test whether a knot complement contains a closed essential surface. This property has important theoretical and algorithmic consequences. However systematically testing it has until now been infeasibly slow, and current techniques only apply to specific families of knots. As a testament to its practicality, we run the algorithm over a comprehensive body of 2979 knots, including the two 20-crossing dodecahedral knots, yielding results that were not previously known.The algorithm derives from the original Jaco-Oertel framework, involves both enumeration and optimisation procedures, and combines several techniques from normal surface theory. This represents substantial progress in the practical implementation of normal surface theory. Problems of this kind have a doubly-exponential time complexity; nevertheless, with our new algorithm we are able to solve it for a large and comprehensive class of inputs. Our methods are relevant for other difficult computational problems in 3-manifold theory, ranging from testing for Haken-ness to the recognition problem for knots, links and 3-manifolds.

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Combinatorial Cubings, Cusps, and the Dodecahedral Knots

Cited by 30 publications

References 4 publications

Commensurability classes of 2–bridge knot complements

Commensurability classes of 2–bridge knot complements

Commensurators of Cusped Hyperbolic Manifolds

Computing closed essential surfaces in knot complements

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