An estimate of the Rasmussen invariant for links and the determination for certain links

Kawamura, Tomomi

doi:10.1016/j.topol.2015.05.034

Cited by 11 publications

(16 citation statements)

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“…Remark 2. A special case of (Cbound) without the δ − -term has been proved for nonpositive and non-negative non-split link diagrams by Kawamura [12]. However, we remark that Kawamura's bound needs the non-split and the non-positivity/non-negativity hypotheses, while our bound does not.…”

Section: Introductionmentioning

confidence: 70%

A Bennequin-Type Inequality and Combinatorial Bounds

Collari¹

2019

Michigan Math. J.

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In this paper we provide a new Bennequin-type inequality for the Rasmussen-Beliakova-Wehrli invariant, featuring the numerical transverse braid invariants (the cinvariants) introduced by the author. From the Bennequin type-inequality, and a combinatorial bound on the value of the c-invariants, we deduce a new computable bound on the Rasmussen invariant. arXiv:1707.03424v3 [math.GT]Theorem 2 (Bennequin-type inequality). Let λ be an oriented link type, and let T be a transverse representative of λ. Then, for each field F, the following inequality holdswhere s(λ; F) is the Rasmussen-Beliakova-Werhli (RBW) s-invariant of λ ([22, 5]).The inequality in Theorem 2 is part of a family of inequalities relating topological and transverse invariants. These bounds are named collectively Bennequin-type inequalities. In particular, our inequality sharpens a similar result due, independently, to O.

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Section: Introductionmentioning

confidence: 70%

A Bennequin-Type Inequality and Combinatorial Bounds

Collari¹

2019

Michigan Math. J.

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“…The Rasmussen invariant of a classical knot extracts geometric information from Khovanov homology, yielding a lower bound on the slice genus [26]. Given a classical knot K it is, in principle, difficult to compute its Rasmussen invariant, denoted s(K), as it is equivalent to the maximal filtration grading of all elements homologous to a certain generator of the Lee homology of K. Kawamura [20] and Lobb [22] independently defined diagram-dependent upper bounds on s(K), denoted U (D) (for D a diagram of K), which are easily computable by hand, along with an error term, ∆(D), the vanishing of which implies that s(K) = U (D), in fact. More precisely,…”

Section: The Slice-bennequin Boundsmentioning

confidence: 99%

Computations of the slice genus of virtual knots

Rushworth¹

2019

Topology and its Applications

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This paper subsumes the (now withdrawn) arXiv submission On the virtual Rasmussen invariant.Abstract. A virtual knot is an equivalence class of embeddings of S 1 into thickened (closed oriented) surfaces, up to self-diffeomorphism of the surface and certain handle stabilisations. The slice genus of a virtual knot is defined diagrammatically, in direct analogy to that of a classical knot. However, it may be defined, equivalently, as follows: a representative of a virtual knot is an embedding of S 1 into a thickened surface Σg ×I; what is the minimal genus of oriented surfaces S → M × I with the embedded S 1 as boundary, where M is an oriented 3-manifold with ∂M = Σg?We compute and estimate the slice genus of all virtual knots of 4 classical crossings or less. We also compute or estimate the slice genus of 46 virtual knots of 5 and 6 classical crossings whose slice status is not determined in the work of Boden, Chrisman, and Gaudreau. The computations are made using two distinct virtual extensions of the Rasmussen invariant, one due to Dye, Kaestner, and Kauffman, the other due to the author. Specifically, the computations are made using bounds on the two extensions of the Ramussen invariant which we construct and investigate. The bounds are themselves generalisations of those on the classical Rasmussen invariant due, independently, to Kawamura and Lobb. The bounds allow for the computation of the extensions of the Rasmussen invariant in particular cases. As asides we identify a class of virtual knots for which the two extensions of the Rasmussen invariant agree, and show that the extension due to Dye, Kaestner, and Kauffman is additive with respect to the connect sum.

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“…The s invariants of general pretzel knots with only one negatively twisted strand were computed by Kawamura [6], which are turned out to be equal to −σ and 2τ . The authors [8] also determined s invariants of an infinite family of pretzel knots other than Kawamura's examples, which are equal to −σ and 2τ .…”

Section: Computationsmentioning

confidence: 99%

“…Note that quasi-alternating pretzel knots are Khovanov homologically σ-thin [12] and are classified by Greene [4]. Kawamura [6] gave an estimate of Rasmussen invariant and determined s invariants of general pretzel knots with only one negatively twisted strand, in which case s are equal to −σ and the twice values of Ozsváth-Szábo Heegaard Floer τ invariant. The authors [8] combined known crossing change formulas to compute s invariants of a family of general preztel knots with many negatively twisted strands, which are again equal to −σ and 2τ .…”

Section: Introductionmentioning

confidence: 99%