Bipyramid Decompositions of Multi-crossing Link Complements

Adams, Colin; Kehne, Gregory

doi:10.48550/arxiv.1610.03830

Cited by 3 publications

(4 citation statements)

References 11 publications

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“…This decomposition was used in [13] to triangulate the infinite square weave. It was extended to alternating links in S 3 in [4] and to links in thickened higher genus surfaces in [3]. Our work here for links in T 2 × I was done independently.…”

Section: Torihedramentioning

confidence: 99%

Geometry of biperiodic alternating links

Champanerkar

Kofman

Purcell

2018

J. London Math. Soc.

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A biperiodic alternating link has an alternating quotient link in the thickened torus. In this paper, we focus on semi‐regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a corresponding Euclidean tiling. Consequently, we determine the exact volumes of semi‐regular links. We relate their commensurability and arithmeticity to the corresponding tiling, and assuming a conjecture of Milnor, we show there exist infinitely many pairwise incommensurable semi‐regular links with the same invariant trace field. We show that only two semi‐regular links have totally geodesic checkerboard surfaces; these two links satisfy the Volume Density Conjecture. Finally, we give conditions implying that many additional biperiodic alternating links are hyperbolic and admit a positively oriented, unimodular geometric triangulation. We also provide sharp upper and lower volume bounds for these links.

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Section: Torihedramentioning

confidence: 99%

Geometry of biperiodic alternating links

Champanerkar

Kofman

Purcell

2018

J. London Math. Soc.

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“…These experiments have been repeated by Adams and Kehne [Keh16]. They have also proved that the expected volume is at most 4π n log n, by constructing a pyramid decomposotion of the petal knot complement [Ada17,AK16]. Any such lower bound would be of great interest.…”

Section: Hyperbolic Volumementioning

confidence: 99%

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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“…Experiments by the authors and by Adams and Kehne [AK16,Keh16] indicate that the hyperbolic volume of K 2n+1 is typically of order n log n, which yields a similar lower bound for the typical c(K 2n+1 ). See [DHO + 14] for experiments on the hyperbolic volume and the number of crossings in a different model.…”

Section: Petal Number and Crossing Numbermentioning

confidence: 75%

“…In fact, the above-mentioned experiments by Adams and Kehne [AK16,Keh16] indicate that K 2n+1 is prime with high probability. Why is this?…”

Section: Theorem 7 ([Erd45]mentioning

confidence: 99%

The distribution of knots in the Petaluma model

Even-Zohar

Hass

Linial

et al. 2018

Algebr. Geom. Topol.

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The representation of knots by petal diagrams (Adams et al. 2012) naturally defines a sequence of distributions on the set of knots. In this article we establish some basic properties of this randomized knot model. We prove that in the random n-petal model the probability of obtaining every specific knot type decays to zero as n, the number of petals, grows. In addition we improve the bounds relating the crossing number and the petal number of a knot. This implies that the n-petal model represents at least exponentially many distinct knots.Past approaches to showing, in some random models, that individual knot types occur with vanishing probability, rely on the prevalence of localized connect summands as the complexity of the knot increases. However this phenomenon is not clear in other models, including petal diagrams, random grid diagrams, and uniform random polygons. Thus we provide a new approach to investigate this question.MSC 57M25, 60B05

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Bipyramid Decompositions of Multi-crossing Link Complements

Cited by 3 publications

References 11 publications

Geometry of biperiodic alternating links

Geometry of biperiodic alternating links

Models of random knots

The distribution of knots in the Petaluma model

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