Fundamentals of knot theory

Kawauchi, Akio

doi:10.1007/978-3-0348-9227-8_1

Cited by 25 publications

(43 citation statements)

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“…[43,44] belongs to family of starfish diagram with three-fingers. We will see that there are plenty of mutants within the starfish family, which belong to the familiar class called pretzel knots [49]. Interestingly, we can calculate R = [21] colored HOMFLY polynomials for such mutant pairs and explicitly check whether the mutants are distinguishable or not.…”

Section: Jhep07(2015)109mentioning

confidence: 99%

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Colored HOMFLY polynomials of knots presented as double fat diagrams

Миронов

Морозов

Ramadevi

et al. 2015

J. High Energ. Phys.

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Many knots and links in S 3 can be drawn as gluing of three manifolds with one or more four-punctured S 2 boundaries. We call these knot diagrams as double fat graphs whose invariants involve only the knowledge of the fusion and the braiding matrices of four -strand braids. Incorporating the properties of four-point conformal blocks in WZNW models, we conjecture colored HOMFLY polynomials for these double fat graphs where the color can be rectangular or non-rectangular representation. With the recent work of Gu-Jockers, the fusion matrices for the non-rectangular [21] representation, the first which involves multiplicity is known. We verify our conjecture by comparing with the [21] colored HOMFLY of many knots, obtained as closure of three braids. The conjectured form is computationally very effective leading to writing [21]-colored HOMFLY polynomials for many pretzel type knots and non-pretzel type knots. In particular, we find class of pretzel mutants which are distinguished and another class of mutants which cannot be distinguished by [21] representation. The difference between the [21]-colored HOMFLY of two mutants seems to have a general form, with A-dependence completely defined by the old conjecture due to Morton and Cromwell. In particular, we check it for an entire multi-parametric family of mutant knots evaluated using evolution method.

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Section: Jhep07(2015)109mentioning

confidence: 99%

“…An important application of this identity is the possibility of shifting any pretzel finger of length 1 to any position [49] at any representation R, though permutations of arbitrary fingers are not a symmetry of the knot.…”

Section: Jhep07(2015)109mentioning

confidence: 99%

Colored HOMFLY polynomials of knots presented as double fat diagrams

Миронов

Морозов

Ramadevi

et al. 2015

J. High Energ. Phys.

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“…It is defined as the signature of any symmetrized Seifert matrix of a given link (see [7]). We may define a signature for braids, as well, via their natural closure in R 3 .…”

Section: Figure 4 Cobordism S(α β)mentioning

confidence: 99%

Asymptotic Rasmussen invariant

Baader

2007

Comptes Rendus. Mathématique

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Abstract. We use simple properties of the Rasmussen invariant of knots to study its asymptotic behaviour on the orbits of a smooth volume preserving vector field on a compact domain in the 3-space. A comparison with the asymptotic signature allows us to prove that asymptotic knots are non-alternating, in general. Further we show that the Rasmussen invariant defines a quasi-morphism on the braid groups and derive estimates for the stable commutator and torsion lengths of alternating braids.

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“…In co-dimension 2 knot theory [6], typically the term ' -knot' denotes a manifold pair ( +2 , ) where is the image of a smooth embedding : → +2 . An -ball pair is a pair ( +2 , ) where is the image of a smooth embedding : → +2 such that −1 (∂ +2 ) = ∂ .…”

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confidence: 99%