Khovanov Homology for Virtual Knots With Arbitrary Coefficients

Manturov, Vassily Olegovich

doi:10.1142/s0218216507005336

Cited by 51 publications

(94 citation statements)

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“…In [8] we have investigated this categorification and other related ones for the arrow polynomial. So far this categorification is effective in that we [31] structured along the lines proposed by Manturov [44], and we expect all this work to reflect back on better understanding of Khovanov homology for classical knots. This is the first instance of non-trivial use of categorified homology in virtual knot theory.…”

Section: Dotted Gradings and The Dotted Categorificationmentioning

confidence: 97%

Introduction to Virtual Knot Theory

Kauffman

2012

J. Knot Theory Ramifications

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This paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial and categorifications of the arrow polynomial.This paper is relatively self-contained, with Sec. 2 reviewing the definitions of virtual knots and links, Sec. 3 reviewing the surface interpretations of virtuals and Sec. 4 reviewing the definitions and properties of flat virtuals.Section 5 proves that long flat virtual knots (virtual strings in Turaev's terminology) embed into long virtual knots. This result is a very strong tool for discriminating long virtual knots from one another. We give examples and computations using this technique by applying invariants of virtual knots to corresponding ascending diagram images of flat virtual knots. A long virtual can be closed just as a long knot can be closed. It is a remarkable property of long virtuals (both flat and not flat) that there are non-trivial long examples whose closures are trivial. Thus one would like to understand the kernel of the closure mapping from long virtuals to virtuals both in the flat and non-flat categories. It is hoped that the invariants discussed in this paper will further this question. We also introduce in Sec. 5 the notion of the parity (odd or even) of a crossing in a virtual knot and recall the odd writhe of a virtual knot -an invariant that is a kind of self-linking number for virtual knots.Section 6 is a review of the bracket polynomial and the Jones polynomial for virtual knots and links. We recall the virtualization construction that produces infinitely many non-trivial virtual knots with unit Jones polynomial. These examples are of interest since there are no known examples of classical non-trivial knots with unit Jones polynomial. One may conjecture that all non-trivial examples produced by virtualization are non-classical. It may be that the virtualization of some non-trivial classical knot is isotopic to a classical knot, but we have no evidence that this can happen. The arrow polynomial invariant developed in this paper may help in deciding this question of non-classicality.Section 7 introduces the Parity Bracket, a modification of the bracket polynomial due to Vassily Manturov that takes into account the parity of crossings in a knot diagram. The Parity Bracket takes values in sums of certain reduced graphs with polynomial coefficients. We show the power of this invariant by given an example of a virtual knot with unit Jones polynomial that is not obtained from a classical knot by the virtualization construction of the previous section. This is the first time such an example has been pinpointed and we do not know any method to detect such examples at this time other than the Parity Bracket of Manturov.In Sec. 8 we give the definition of the arrow polynomial A[K] and a number of examples of its calculation. These examples include a verification of the nonclassicality of the simplest example of virtualization, a verification that the Kish...

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Section: Dotted Gradings and The Dotted Categorificationmentioning

confidence: 97%

Introduction to Virtual Knot Theory

Kauffman

2012

J. Knot Theory Ramifications

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“…The subsequent section we consider how the cocycle can be used for reformulation of some invariants of virtual links in a form which does not account virtual crossing. We conclude the paper with a modification of the construction of Khovanov homology for virtual links [4,19]; the modification exploits the parity cocycle, i.e. the index cocycle mod 2.…”

Section: Figure 5: Reidemeister Moves On Gauss Diagramsmentioning

confidence: 99%

“…As an application of the parity cocycle we consider a construction of the Khovanov homology of virtual links which uses a local source-sink structure. The construction below is a reformulation of the Khovanov homology constructions in [4,19,20].…”

Section: Khovanov Homology Of Virtual Linksmentioning

confidence: 99%

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Virtual index cocycles and invariants of virtual links

Nikonov¹

2020

Preprint

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“…For some different constructions of link invariants in manifolds M 2 × R 1 and closely related virtual link invariants, mainly developing the Kauffman and Khovanov constructions, see [18], [19] and references therein, and also [10]. + and s ′ − the results of our isotopy J applied to s + and s − respectively, in particular s ′…”

Section: Remarkmentioning

confidence: 99%