Universal knot diagrams

Even-Zohar, Chaim; Hass, Joel; Linial, Nathan; Nowik, Tahl

doi:10.1142/s0218216519500317

Cited by 12 publications

(6 citation statements)

References 43 publications

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“…Note that one link can be in several such sets for different r, s. Proof. This follows from Theorem 2.2 together with Theorem 1.4 of [7]. Theorem 2.3 means, in particular, that the set of all links is the same as the set of (r, 2s − 1)-meander links for all natural r and s, up to link isotopy.…”

Section: Theorem 22 Every Knot Has a Meander Diagrammentioning

confidence: 87%

Random meander model for links

Owad¹,

Tsvietkova²

2022

Preprint

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We describe a new random model for links based on meanders. Random meander diagrams correspond to matching pairs of parentheses, a well-studied problem in combinatorics. Hence tools from combinatorics can be used to investigate properties of links. We prove that unlinks appear with vanishing probability, no link L is obtained with probability 1, and there is a lower bound for the number of non-isotopic knots obtained on every step. Then we give expected twist number of a diagram, and bound expected hyperbolic and simplicial volume of links.

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Section: Theorem 22 Every Knot Has a Meander Diagrammentioning

confidence: 87%

Random meander model for links

Owad¹,

Tsvietkova²

2022

Preprint

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“…In this section, we introduce some basic facts on lattice diagrams to rely on in what follows. One can see recent paper [23] for detailed explanations.…”

Section: Basic Theoremsmentioning

confidence: 99%

“…In [23], lattice diagrams are called potholder curves or potholders. Knots carried by potholder curves were studied by Grosberg and Nechaev [33,34], who calculated the number of unknots carried by such curves via a connection to the Potts model of statistical mechanics.…”

Section: Basic Theoremsmentioning

confidence: 99%

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On an alternative stratification of knots

Lanina¹,

Popolitov²,

Tselousov³

2022

Preprint

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We introduce an alternative stratification of knots: by the size of lattice on which a knot can be first met. Using this classification, we find ratio of unknots and knots with more than 10 minimal crossings inside different lattices and answer the question which knots can be realized inside 3 × 3 and 5 × 5 lattices. In accordance with previous research, the ratio of unknots decreases exponentially with the growth of the lattice size. Our computational results are approved with theoretical estimates for amounts of knots with fixed crossing number lying inside lattices of given size.

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“…The existence of such diagrams was shown by Adams, Shinjo, and Tanaka, using meander diagrams for knots [AST11]. These results were extended by Even-Zohar, Hass, Linial and Nowik to links with an arbitrary number of components, using potholder diagrams [EHLN19]. The proofs in these two works are constructive, providing algorithms that compute a new diagram of a given link L, with at most two odd-sided faces.…”

mentioning

confidence: 92%

Random Colorings in Manifolds

Even-Zohar¹,

Hass²

2022

Preprint

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We develop a general method for constructing random manifolds and submanifolds in arbitrary dimensions. The method is based on associating colors to the vertices of a triangulated manifold, as in recent work for curves in 3-dimensional space by Sheffield and Yadin (2014). We determine conditions on which submanifolds can arise, in terms of Stiefel-Whitney classes and other properties. We then consider the random submanifolds that arise from randomly coloring the vertices. Since this model generates submanifolds, it allows for studying properties and using tools that are not available in processes that produce general random subcomplexes. The case of 3 colors in a triangulated 3-ball gives rise to random knots and links. In this setting, we answer a question raised by de Crouy-Chanel and Simon (2019), showing that the probability of generating an unknot decays exponentially. In the general case of k colors in d-dimensional manifolds, we investigate the random submanifolds of different codimensions, as the number of vertices in the triangulation grows. We compute the expected Euler characteristic, and discuss relations to homological percolation and other topological properties. Finally, we explore a method to search for solutions to topological problems by generating random submanifolds. We describe computer experiments that search for a low-genus surface in the 4-dimensional ball whose boundary is a given knot in the 3-dimensional sphere.

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Universal knot diagrams

Cited by 12 publications

References 43 publications

Random meander model for links

Random meander model for links

On an alternative stratification of knots

Random Colorings in Manifolds

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