A Tabulation of Oriented Links

Doll, Helmut; Hoste, Jim

doi:10.2307/2938715

Cited by 8 publications

(7 citation statements)

References 8 publications

(15 reference statements)

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“…Proof. Let ∆ D (x) be the Alexander polynomial of D, then ∆ D (x) = P (D, x 1/2 − x −1/2 , 1) [5]. Assume that D is positive leading, substituting a = 1, z = x 1/2 − x −1/2 in (3.1) leads to…”

Section: The Braid Index Of Reduced Alternating Braidsmentioning

confidence: 99%

The HOMFLY polynomial of links in closed braid form

Liu

Diao

Hetyei

2019

Discrete Mathematics

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It is well known that any link can be represented by the closure of a braid. The minimum number of strings needed in a braid whose closure represents a given link is called the braid index of the link and the well known Morton-Frank-Williams inequality reveals a close relationship between the HOMFLY polynomial of a link and its braid index. In the case that a link is already presented in a closed braid form, Jaeger derived a special formulation of the HOMFLY polynomial. In this paper, we prove a variant of Jaeger's result as well as a dual version of it. Unlike Jaeger's original reasoning, which relies on representation theory, our proof uses only elementary geometric and combinatorial observations. Using our variant and its dual version, we provide a direct and elementary proof of the fact that the braid index of a link that has an n-string closed braid diagram that is also reduced and alternating, is exactly n. Until know this fact was only known as a consequence of a result due to Murasugi on fibered links that are star products of elementary torus links and of the fact that alternating braids are fibered.

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Section: The Braid Index Of Reduced Alternating Braidsmentioning

confidence: 99%

The HOMFLY polynomial of links in closed braid form

Liu

Diao

Hetyei

2019

Discrete Mathematics

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“…We use the notation of Doll and Hoste to differentiate isotopy classes of the same link type [9]. [8,10].…”

Section: Doll and Hoste Notationmentioning

confidence: 99%

“…Example of the link notation adopted and modified from the work of Doll and Hoste[9]. The lighter blue strand is component 1 and the darker red-orange strand is component 2.…”

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confidence: 99%

A Symmetry Motivated Link Table

Witte¹,

Flanner²,

Vázquez³

2018

Preprint

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Proper identification of oriented knots and 2-component links requires a precise link nomenclature. Motivated by questions arising in DNA topology, this study aims to produce a nomenclature unambiguous with respect to link symmetries. For knots, this involves distinguishing a knot type from its mirror image. In the case of 2-component links, there are up to sixteen possible symmetry types for each topology. The study revisits the methods previously used to disambiguate chiral knots and extends them to oriented 2-component links with up to nine crossings. Monte Carlo simulations are used to report on writhe, a geometric indicator of chirality. There are ninety-two prime 2-component links with up to nine crossings. Guided by geometrical data, linking number and the symmetry groups of 2-component links, a canonical link diagram for each link type is proposed. All diagrams but six were unambiguously chosen ( 8 1 2 5 , 9 5 2 , 9 3 2 4 , 9 3 2 5 , 9 3 2 9 , and 9 4 2 1 ). We include complete tables for prime knots with up to ten crossings and prime links with up to nine crossings. We also prove a result on the behavior of the writhe under local lattice moves.

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“…(By the term 'knot', we intend a link of one component.) We use the well-known notation of Alexander and Briggs (1926) [3], the data being abstracted from [1], itself citing [36] and [8] (beware that the tables in this latter article are presented in microfiche form only).…”

Section: Link Examplesmentioning

confidence: 99%

On the Links–Gould Invariant of Links

Wit

Links

Kauffman

1999

J. Knot Theory Ramifications

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We introduce and study in detail an invariant of (1, 1) tangles. This invariant, derived from a family of four dimensional representations of the quantum superalgebra U q [gl(2|1)], will be referred to as the Links-Gould invariant. We find that our invariant is distinct from the Jones, HOMFLY and Kauffman polynomials (detecting chirality of some links where these invariants fail), and that it does not distinguish mutants or inverses. The method of evaluation is based on an abstract tensor state model for the invariant that is quite useful for computation as well as theoretical exploration.

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A Tabulation of Oriented Links

Cited by 8 publications

References 8 publications

The HOMFLY polynomial of links in closed braid form

The HOMFLY polynomial of links in closed braid form

A Symmetry Motivated Link Table

On the Links–Gould Invariant of Links

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