Unsolved problems in virtual knot theory and combinatorial knot theory

Fenn, Roger; Ilyutko, Denis Petrovich; Kauffman, Louis H.; Manturov, Vassily Olegovich

doi:10.4064/bc103-0-1

Cited by 15 publications

(13 citation statements)

References 160 publications

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“…Our construction of Khovanov homology for virtual knots is a reformulation of the theory developed by Vassily Manturov in [34]. The reader interested in seeing more background about virtual knot theory can consult [10,22,36].…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

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confidence: 99%

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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: Introductionmentioning

confidence: 99%

mentioning

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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“…Some generalizations of the Alexander polynomial for virtual knots can be found in [45] and [47], and some generalizations of the Jones polynomial can be found in [40,41] and [12]. Readers should refer to [15] for some recent progress and open problems in virtual knot theory.…”

Section: Theorem 22 ([17]) a Gauss Diagram Uniquely Defines A Virtumentioning

confidence: 99%

The Chord Index, its Definitions, Applications, and Generalizations

Cheng

2020

Can. J. Math.-J. Can. Math.

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In this paper we study the chord index of virtual knots, which can be thought of as an extension of the chord parity. We show how to use the chord index to define finite type invariants of virtual knots. The notions of indexed Jones polynomial and indexed quandle are introduced, which generalize the classical Jones polynomial and knot quandle respectively. Some applications of these new invariants are discussed. We also study how to define a generalized chord index via a fixed finite biquandle. Finally the chord index and its applications in twisted knot theory are discussed.1991 Mathematics Subject Classification. 57M25, 57M27.

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“…By virtualizing these crossings while leaving the rest of the diagram just as before, we can obtain a graphical virtual knot diagram with unit Jones polynomial which is non-trivial and non-classical [9]. We refer the reader to examine [9,15,17,7] for the details of this construction. Proof.…”

Section: Examplesmentioning

confidence: 99%

Graphical virtual links and a polynomial for signed cyclic graphs

Deng

Kauffman

2018

J. Knot Theory Ramifications

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For a signed cyclic graph G, we can construct a unique virtual link L by taking the medial construction and convert 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In the article we shall prove that a virtual link L is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial F [G] for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between F [G] of a signed cyclic graph G and the bracket polynomial of one of the virtual link diagrams associated with G. Finally we give a spanning subgraph expansion for F [G].

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Unsolved problems in virtual knot theory and combinatorial knot theory

Abstract: This paper is a concise introduction to virtual knot theory, coupled with a list of research problems in this field.

Cited by 15 publications

References 160 publications

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

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

The Chord Index, its Definitions, Applications, and Generalizations

Graphical virtual links and a polynomial for signed cyclic graphs

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