Virtual Knot Cobordism and Bounding the Slice Genus

Bodén, Hans; Chrisman, Micah; Gaudreau, Robin

doi:10.1080/10586458.2017.1422160

Cited by 15 publications

(24 citation statements)

References 25 publications

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“…They include the index polynomial of Heinrich [14] and the graded genus of Turaev [29]. Boden, Chrisman, and Gaudreau [4] have used these invariants and others to compute or estimate the slice genus of a very large number of the 92800 virtual knots of 6 crossing or less (as given in Green's table [13]).…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…Exact values of g * are obtained by constructing a cobordism which attains a lower bound given by s, s 1 , or s 2 . Upper bounds on g * are obtained by constructing a cobordism of the given genus, and employing the fact that half the crossing number bounds the slice genus of a knot from above (as in the classical case) [4]. Shortly after posting a previous version of this paper to the arXiv the author learned of the work of Boden, Chrisman, and Gaudreau in which they compute or estimate the slice genus of a very large number of the virtual knots of 6 crossings or less [4,5].…”

Section: Computation and Estimation Of The Slice Genusmentioning

confidence: 99%

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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This paper subsumes the (now withdrawn) arXiv submission On the virtual Rasmussen invariant.Abstract.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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Section: Statement Of Resultsmentioning

confidence: 99%

Section: Computation and Estimation Of The Slice Genusmentioning

confidence: 99%

Computations of the slice genus of virtual knots

Rushworth¹

2019

Topology and its Applications

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“…This paper studies the concordance invariance of the affine index polynomial [19], denoted P K (t). This invariant is also called the writhe polynomial, W K (t), in the context of Gauss diagrams, see [1] where the W notation is used and where a related polynomial is denoted by P K (t) (there should be no confusion). In this paper we work in the context of virtual knot and link diagrams and extend the definitions in [19] to an affine index polynomial for links (with affine labeling as described in the body of the paper).…”

Section: Introductionmentioning

confidence: 99%

“…In Section 4 we re-develop the affine index polynomial [19] extending its definition to labeled links and proving that it is a concordance invariant of virtual knots and links. The reader should note that concordance invariance of the affine index polynomial for knots is proved in [1] by Gauss diagram techniques. In the present work we use affine labelings of the knot and link diagrams.…”

Section: Introductionmentioning

confidence: 99%

Virtual knot cobordism and the affine index polynomial

Kauffman

2018

J. Knot Theory Ramifications

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This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant under certain forms of labeled cobordism and study a number of examples in relation to these phenomena. Information on determinations of the four-ball genus of some virtual knots is obtained by via the affine index polynomial in conjunction with results on the genus of positive virtual knots using joint work with Dye and Kaestner.

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“…Here we consider two related questions: (1) how does one draw a Seifert surface for a homologically trivial knot in a thickened surface Σ × [0, 1], where Σ is compact, connected, and oriented, and (2) how can such surfaces be employed in the study of virtual knots? These issues arise, for example, in the computation of the directed signature functions [1] of Boden, Gaudreau, and the author.…”

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confidence: 99%

Virtual Seifert surfaces

Chrisman

2019

J. Knot Theory Ramifications

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A virtual knot that has a homologically trivial representative K in a thickened surface Σ × [0, 1] is said to be an almost classical (AC) knot. K then bounds a Seifert surface F ⊂ Σ × [0, 1]. Seifert surfaces of AC knots are useful for computing concordance invariants and slice obstructions. However, Seifert surfaces in Σ × [0, 1] are difficult to construct. Here we introduce virtual Seifert surfaces of AC knots. These are planar figures representing F ⊂ Σ × [0, 1]. An algorithm for constructing a virtual Seifert surface from a Gauss diagram is given. This is applied to computing signatures and Alexander polynomials of AC knots. A canonical genus of AC knots is also studied. It is shown to be distinct from the virtual canonical genus of Stoimenow-Tchernov-Vdovina.

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Virtual Knot Cobordism and Bounding the Slice Genus

Cited by 15 publications

References 25 publications

Computations of the slice genus of virtual knots

Computations of the slice genus of virtual knots

Virtual knot cobordism and the affine index polynomial

Virtual Seifert surfaces

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