A Linear Time Algorithm for Embedding Arbitrary Knotted Graphs into a 3-Page Book

Kurlin, Vitaliy; Smithers, Christopher J.

doi:10.1007/978-3-319-29971-6_6

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Introduction: Motivations and The Realizability Problem1

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2020

2021

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Cited by 2 publications

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“…Other approaches to knotted structures are 3-page embeddings that encode isotopy classes of spatial graphs by central elements of finitely presented semigroups [8], [13], [9], [12].…”

Section: Introduction: Motivations and The Realizability Problemmentioning

confidence: 99%

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Encoding and topological computation on textile structures

Bright

Kurlin

2020

Computers & Graphics

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A textile structure is a periodic arrangement of threads in the thickened plane. A topological classification of textile structures is harder than for classical knots and links that are non-periodic and restricted to a bounded region. The first important problem is to encode all textile structures in a simple combinatorial way. This paper extends the notion of the Gauss code in classical knot theory, providing a tool for topological computation on these structures. As a first application, we present a linear time algorithm for determining whether a code represents a textile in the physical sense. This algorithm, along with invariants of textile structures, allowed us for the first time to classify all oriented textile structures woven from a single component up to complexity five.

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“…Other approaches to knotted structures are 3-page embeddings that encode isotopy classes of spatial graphs by central elements of finitely presented semigroups [8], [13], [9], [12].…”

Section: Introduction: Motivations and The Realizability Problemmentioning

confidence: 99%

“…10,11,12 explain the geometric meaning of choosing the next oriented edge by turning left in a torus diagram represented by the code W.…”

mentioning

confidence: 99%