A family of transversely nonsimple knots

Khandhawit, Tirasan; Ng, Lenhard

doi:10.2140/agt.2010.10.293

Cited by 18 publications

(25 citation statements)

References 23 publications

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“…The d t invariant of the braid A(0, 0) := σ 3 σ −2 2 σ 2 3 σ 2 σ −1 3 σ −1 1 σ 2 σ 2 1 (notation from [25] and [22]) is given below. The profile drawn above is based on the computations of d t at values t = k 24 , k = 1, .…”

Section: Examplesmentioning

confidence: 99%

Annular Khovanov–Lee homology, braids, and cobordisms

Grigsby

Licata

Wehrli

2017

Pure and Applied Mathematics Quarterly

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We prove that the Khovanov-Lee complex of an oriented link, L, in a thickened annulus, A × I, has the structure of a (Z ⊕ Z)-filtered complex whose filtered chain homotopy type is an invariant of the isotopy class of L ⊂ (A × I). Using ideas of Ozsváth-Stipsicz-Szabó [33] as reinterpreted by Livingston [30], we use this structure to define a family of annular Rasmussen invariants that yield information about annular and non-annular cobordisms. Focusing on the special case of annular links obtained as braid closures, we use the behavior of the annular Rasmussen invariants to obtain a necessary condition for braid quasipositivity and a sufficient condition for right-veeringness. ∂ = ∂ 0 + ∂ − and thus endow the Khovanov complex with the structure of a Z-filtered complex, a structure that has proven to be particularly well-suited for studying braids and their conjugacy

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Section: Examplesmentioning

confidence: 99%

Annular Khovanov–Lee homology, braids, and cobordisms

Grigsby

Licata

Wehrli

2017

Pure and Applied Mathematics Quarterly

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“…It is clear that the projection and inclusion maps j The map W .R 3 ; rot / ! .R 3 ; std / defined by .x; y; z/ D .x; 2y; xy C z/ is a contactomorphism [19]. A result of Bennequin states that 1 .K/ is transversely isotopic in .R 3 ; rot / to a transverse braid Tˇabout the z -axis [6], as shown in Figure 25.…”

Section: Remark 61mentioning

confidence: 99%

On the equivalence of Legendrian and transverse invariants in knot Floer homology

2013

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Using the grid diagram formulation of knot Floer homology, Ozsváth, Szabó and Thurston defined an invariant of transverse knots in the tight contact 3-sphere. Shortly afterwards, Lisca, Ozsváth, Stipsicz and Szabó defined an invariant of transverse knots in arbitrary contact 3-manifolds using open book decompositions. It has been conjectured that these invariants agree where they are both defined. We prove this fact by defining yet another invariant of transverse knots, showing that this third invariant agrees with the two mentioned above.

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“…It is a simple exercise, see for instance [15,Figure 6], to show that the horizontal green and the blue curve in Figure 6 map to the large green and blue curve in Figure 1. Similarly the vertical green curve, perturbed not to go through infinity, in Figure 6 maps to the other green curve in Figure 1.…”

Section: Transverse Universal Linksmentioning

confidence: 99%