Knots and links

Negami found an upper bound on the stick number [Formula: see text] of a nontrivial knot [Formula: see text] in terms of the minimal crossing number [Formula: see text]: [Formula: see text]. Huh and Oh found an improved upper bound: [Formula: see text]. Huh, No and Oh proved that [Formula: see text] for a [Formula: see text]-bridge knot or link [Formula: see text] with at least six crossings. As a sequel to this study, we present an upper bound on the stick number of Montesinos knots and links. Let [Formula: see text] be a knot or link which admits a reduced Montesinos diagram with [Formula: see text] crossings. If each rational tangle in the diagram has five or more index of the related Conway notation, then [Formula: see text]. Furthermore, if [Formula: see text] is alternating, then we can additionally reduce the upper bound by [Formula: see text].

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Piecewise-linear embeddings of knots and links with rotoinversion symmetry

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273321006136 ◽

2021 ◽

Vol 77 (5) ◽

Author(s):

Michael O'Keeffe ◽

Michael M. J. Treacy

Keyword(s):

Piecewise Linear ◽

Infinite Family ◽

Knots And Links

This article describes the simplest members of an infinite family of knots and links that have achiral piecewise-linear embeddings in which linear segments (sticks) meet at corners. The structures described are all corner- and stick-2-transitive – the smallest possible for achiral knots.

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Biquasiles and dual graph diagrams

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216517500481 ◽

2017 ◽

Vol 26 (08) ◽

pp. 1750048 ◽

Cited By ~ 8

Author(s):

Deanna Needell ◽

Sam Nelson

Keyword(s):

Dual Graph ◽

Algebraic Structures ◽

Combinatorial Structures ◽

Reidemeister Moves ◽

Knots And Links ◽

The Way

We introduce dual graph diagrams representing oriented knots and links. We use these combinatorial structures to define corresponding algebraic structures which we call biquasiles whose axioms are motivated by dual graph Reidemeister moves, generalizing the Dehn presentation of the knot group analogously to the way quandles and biquandles generalize the Wirtinger presentation. We use these structures to define invariants of oriented knots and links and provide examples.

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Incoherent nullification of torus knots and links

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216519500330 ◽

2019 ◽

Vol 28 (05) ◽

pp. 1950033

Author(s):

Zac Bettersworth ◽

Claus Ernst

Keyword(s):

Upper Bound ◽

Torus Knots ◽

Number Formula ◽

Knots And Links

In the paper, we study the incoherent nullification number [Formula: see text] of knots and links. We establish an upper bound on the incoherent nullification number of torus knots and links and conjecture that this upper bound is the actual incoherent nullification number of this family. Finally, we establish the actual incoherent nullification number of particular subfamilies of torus knots and links.

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Nullification of Torus knots and links

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216514500588 ◽

2014 ◽

Vol 23 (11) ◽

pp. 1450058 ◽

Cited By ~ 4

Author(s):

Claus Ernst ◽

Anthony Montemayor

Keyword(s):

Torus Knots ◽

The Family ◽

Minimum Number ◽

Knots And Links

It is known that a knot/link can be nullified, i.e. can be made into the trivial knot/link, by smoothing some crossings in a projection diagram of the knot/link. The minimum number of such crossings to be smoothed in order to nullify the knot/link is called the nullification number. In this paper we investigate the nullification numbers of a particular knot family, namely the family of torus knots and links.

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