Span of the Jones polynomial of an alternating virtual link

Kamada, Naoko

doi:10.2140/agt.2004.4.1083

Cited by 30 publications

(32 citation statements)

References 10 publications

(32 reference statements)

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“…Source-sink orientations were originally introduced by Naoko Kamada [15] [16] and determine a version of checkerboard coloring for virtual knots. We note that the source-sink orientations we use can be regarded as a translation of the oriented chord diagrams of Oleg Viro [43].…”

Section: Source-sink Orientations and Cut Locimentioning

confidence: 99%

Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots

Dye

Kaestner

Kauffman

2017

J. Knot Theory Ramifications

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The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov-Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies as well to classical knot and classical knot cobordisms. We give an alternate formulation for the Manturov definition [34] of Khovanov homology [25] [26] for virtual knots and links with arbitrary coefficients. This approach uses cut loci on the knot diagram to induce a conjugation operator in the Frobenius algebra. We use this to show that a large class of virtual knots with unit Jones polynomial are nonclassical, proving a conjecture in [20] and [10]. We then discuss the implications of the maps induced in the aforementioned theory to the universal Frobenius algebra [27] for virtual knots. Next we show how one can apply the Karoubi envelope approach of Bar-Natan and Morrison [3] on abstract link diagrams [17] with cross cuts to construct the canonical generators of the Khovanov-Lee homology [30]. Using these canonical generators we derive a generalization of the Rasmussen invariant [39] for virtual knot cobordisms and generalize Rasmussen's result on the slice genus for positive knots to the case of positive virtual knots. It should also be noted that this generalization of the Rasmussen invariant provides an easy to compute obstruction to knot cobordisms in S g × I × I in the sense of Turaev [42].

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Section: Source-sink Orientations and Cut Locimentioning

confidence: 99%

Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots

Dye

Kaestner

Kauffman

2017

J. Knot Theory Ramifications

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“…This notion was introduced by N. Kamada in [21,22], who showed that many classical results on knots and links can be extended to checkerboard colorable virtual links. A similar notion under the name of atom was studied by V. Manturov [30] following A. Fomenko [12].…”

Section: The Theorem From [7]mentioning

confidence: 99%

Generalized duality for graphs on surfaces and the signed Bollobás–Riordan polynomial

Chmutov

2009

Journal of Combinatorial Theory, Series B

117

239

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We generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed Bollobás-Riordan polynomials of dual graphs. This relation unifies various recent results expressing the Jones polynomial of links as specializations of the Bollobás-Riordan polynomials.

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“…mod m almost classical) virtual link diagram of L. H. Dye introduced the notion of a cut point [3], which is an 'unoriented' cut point in our sense. The author [6] generalized the Kauffman-Murasugi-Thistlethwaite theorem ( [11,14,15]) on the span of the Jones polynomial of a classical link to checkerboard colorable and proper virtual links. Using cut points, H. Dye [4] further extended this result to virtual link diagrams that are not checkerboard colorable.…”

Section: Alexander Numberings and Cut Systemsmentioning

confidence: 99%

Cyclic coverings of virtual link diagrams

Kamada

2019

Int. J. Math.

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A virtual link diagram is called mod m almost classical if it admits an Alexander numbering valued in integers modulo m, and a virtual link is called mod m almost classical if it has a mod m almost classical diagram as a representative. In this paper, we introduce a method of constructing a mod m almost classical virtual link diagram from a given virtual link diagram, which we call an m-fold cyclic covering diagram. The main result is that m-fold cyclic covering diagrams obtained from two equivalent virtual link diagrams are equivalent. Thus we have a well-defined map from the set of virtual links to the set of mod m almost classical virtual links. Some applications are also given.

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Span of the Jones polynomial of an alternating virtual link

Cited by 30 publications

References 10 publications

Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots

Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots

Generalized duality for graphs on surfaces and the signed Bollobás–Riordan polynomial

Cyclic coverings of virtual link diagrams

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