Turaev genus and alternating decompositions

Armond, Cody; Lowrance, Adam M.

doi:10.2140/agt.2017.17.793

Cited by 10 publications

(9 citation statements)

References 30 publications

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“…Armond and Lowrance [3] proved a similar classification independently at the same time. They classified link diagrams with Turaev genus one and two in terms of their alternating decomposition graphs upto graph isomorphism.…”

Section: Introductionmentioning

confidence: 67%

Link diagrams with low Turaev genus

Kim

2017

Proc. Amer. Math. Soc.

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We classify link diagrams with Turaev genus one and two in terms of an alternating tangle structure of the link diagram. The proof involves surgery along simple loops on the Turaev surface, called cutting loops, which have corresponding cutting arcs that are visible on the planar link diagram. These also provide new obstructions for a link diagram on a surface to come from the Turaev surface algorithm. We also show that inadequate Turaev genus one links are almost-alternating. arXiv:1507.02918v2 [math.GT] 30 Nov 2015

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Section: Introductionmentioning

confidence: 67%

Link diagrams with low Turaev genus

Kim

2017

Proc. Amer. Math. Soc.

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“…Bloom [Blo10] proved that odd Khovanov homology is mutation invariant. Armond and Lowrance [AL17] proved that if L is a Turaev genus one link, then there is a sequence of mutations transforming L into an almost alternating link. Therefore the desired result holds for Turaev genus one links.…”

Section: 2mentioning

confidence: 99%

Extremal Khovanov homology of Turaev genus one links

Dasbach

Lowrance

2020

Fund. Math.

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The Turaev genus of a link can be thought of as a way of measuring how non-alternating a link is. A link is Turaev genus zero if and only if it is alternating, and in this viewpoint, links with large Turaev genus are very non-alternating. In this paper, we study Turaev genus one links, a class of links which includes almost alternating links. We prove that the Khovanov homology of a Turaev genus one link is isomorphic to Z in at least one of its extremal quantum gradings. As an application, we compute or nearly compute the maximal Thurston Bennequin number of a Turaev genus one link.

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“…If L is a link with g T (L) = 1, then [AL15] implies that L is mutant to an almost alternating link L ′ . Since mutation does not change the Jones polynomial, it follows that V L (t) = V L ′ (t), and the result holds.…”

Section: Jones Polynomialmentioning

confidence: 99%

“…Armond and Lowrance [AL15] and independently Kim [Kim15] classified link diagrams whose Turaev surface is genus one. Every non-split link of Turaev genus one has a diagram obtained by arranging an even number of proper alternating 2-tangles into a circle as in Figure 2.…”

Section: Introductionmentioning

confidence: 99%

Invariants for Turaev genus one links

Dasbach

Lowrance

2018

Communications in Analysis and Geometry

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The Turaev genus defines a natural filtration on knots where Turaev genus zero knots are precisely the alternating knots. We show that the signature of a Turaev genus one knot is determined by the number of components in its all-A Kauffman state, the number of positive crossings, and its determinant. We also show that either the leading or trailing coefficient of the Jones polynomial of a Turaev genus one link (or an almost alternating link) has absolute value one.

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Turaev genus and alternating decompositions

Cited by 10 publications

References 30 publications

Link diagrams with low Turaev genus

Link diagrams with low Turaev genus

Extremal Khovanov homology of Turaev genus one links

Invariants for Turaev genus one links

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