On the Kauffman polynomial of an adequate link

Thistlethwaite, Morwen

doi:10.1007/bf01394334

Cited by 96 publications

(72 citation statements)

References 9 publications

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“…(The Perko knot 10 161 in [25, appendix] is an example; this feature of it transpires from results in [30], where the knot is discussed.) A link is inadequate, if it is neither A-nor B-adequate.…”

Section: Semiadequacy and Kauffman Bracketmentioning

confidence: 92%

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The classification of partially symmetric 3-braid links

Stoimenov

2015

Open Mathematics

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Section: Semiadequacy and Kauffman Bracketmentioning

confidence: 92%

“…[6]). That is, [30], and ultimately [22], we have to invoke another quite substantial body of results. Thus, whether our proof is in the end "simpler" is to some extent a matter of personal view.…”

Section: Oriented Achirality and 3-component Casesmentioning

confidence: 99%

The classification of partially symmetric 3-braid links

Stoimenov

2015

Open Mathematics

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“…The notion of a +adequate link is a common generalization of positive and alternating links, 5 introduced in [7] (see also [14]). In this paper, we use Rutherford's theorem to unify 6 Tanaka's and Ng's approaches: that is, we construct a Legendrian representative with a ruling for each +adequate link.…”

Section: Definitionmentioning

confidence: 99%

“…Thistlethwaite [14] claimed that up to 11 crossings, every knot or its mirror is +adequate. Although Cromwell's example, discussed in Remark 3, refutes this claim, it is probably safe to say that for at least about half of the knots up to 11 crossings, Corollary 6 determines tb.…”

Section: Lemma 5 Let G Be a Planar Graph (Possibly With Multiple Edgmentioning

confidence: 99%

Maximal Thurston--Bennequin number of $+$adequate links

Kálmán

2008

Proc. Amer. Math. Soc.

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Abstract. The class of +adequate links contains both alternating and positive links. Generalizing results of Tanaka (for the positive case) and Ng (for the alternating case), we construct fronts of an arbitrary +adequate link A so that the diagram has a ruling; therefore its Thurston-Bennequin number is maximal among Legendrian representatives of A. We derive consequences for the Kauffman polynomial and Khovanov homology of +adequate links.The maximum Thurston-Bennequin number, denoted by tb, is a knot invariant that has recently drawn a lot of interest. Its definition is possible because Bennequin's inequality, tb ≤ 2g − 1, bounds from above the Thurston-Bennequin number of Legendrian representatives 1 of any knot type by (essentially) the genus g of the knot. Either Bennequin's inequality itself or other bounds, for example the so-called Kauffman bound on tb [10], make the extension to links possible.Recall that the Thurston-Bennequin number is computed from an oriented front diagram by subtracting the number of right cusps from the writhe; tb is the maximum of these numbers for all fronts representing a given link type. The Kauffman bound states that tb, and thus tb, is strictly less than the minimum v-degree (or −1 times the maximum a-degree) of the Kauffman polynomial.

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“…We say that a diagram D is adequate [29] if D is σ-adequate for both the positive state σ + and the negative state σ − . Note also that D is automatically σ ± -homogeneous since σ ± (c j ) = ± for all j. E 2.7.…”

Section: Introductionmentioning

confidence: 99%

Essential State Surfaces for Knots and Links

Ozawa¹

2011

J. Aust. Math. Soc.

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We study a canonical spanning surface obtained from a knot or link diagram, depending on a given Kauffman state. We give a sufficient condition for the surface to be essential. By using the essential surface, we can deduce the triviality and splittability of a knot or link from its diagrams. This has been done on the extended knot or link class that includes all semiadequate, homogeneous knots and links, and most algebraic knots and links. In order to prove the main theorem, we extend Gabai's Murasugi sum theorem to the case of nonorientable spanning surfaces.2010 Mathematics subject classification: primary 57M25; secondary 57Q35.

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On the Kauffman polynomial of an adequate link

Cited by 96 publications

References 9 publications

The classification of partially symmetric 3-braid links

The classification of partially symmetric 3-braid links

Maximal Thurston--Bennequin number of $+$adequate links

Essential State Surfaces for Knots and Links

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