Virtual Khovanov homology using cobordisms

We extend Bar-Natan's cobordism-based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex (h = t = 0), the variant of Lee (h = 0, t = 1) and other classical link homologies. We show that our construction allows, over rings of characteristic two, extensions with no classical analogon, e.g. Bar-Natan's ℤ/2-link homology can be extended in two non-equivalent ways. Our construction is computable in the sense that one can write a computer program to perform calculations, e.g. we have written a MATHEMATICA-based program. Moreover, we give a classification of all unoriented TQFTs which can be used to define virtual link homologies from our topological construction. Furthermore, we prove that our extension is combinatorial and has semi-local properties. We use the semi-local properties to prove an application, i.e. we give a discussion of Lee's degeneration of virtual homology.

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CLASSIFICATION OF CLOSED VIRTUAL 2-BRAIDS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216508006592 ◽

2008 ◽

Vol 17 (10) ◽

pp. 1223-1239 ◽

Cited By ~ 2

Author(s):

TERUHISA KADOKAMI

Keyword(s):

Braid Group ◽

Existence And Uniqueness ◽

Minimal Realization ◽

Virtual Link ◽

Virtual Links ◽

Bracket Polynomial

We classify closed virtual 2-braids completely as virtual links. For the proof, we use surface bracket polynomial due to Dye and Kauffman, Kuperberg's theorem which states existence and uniqueness of a minimal realization for a virtual link, and a subgroup of the 3-braid group.

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Oriented local moves and divisibility of the Jones–Kauffman polynomial

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216519500883 ◽

2019 ◽

Vol 28 (14) ◽

pp. 1950088

Author(s):

Paul Drube ◽

Puttipong Pongtanapaisan

Keyword(s):

Jones Polynomial ◽

Necessary Condition ◽

Virtual Link ◽

Type Formula ◽

Virtual Links ◽

Special Cases ◽

Jones Polynomials ◽

Kauffman Polynomial

For any virtual link [Formula: see text] that may be decomposed into a pair of oriented [Formula: see text]-tangles [Formula: see text] and [Formula: see text], an oriented local move of type [Formula: see text] is a replacement of [Formula: see text] with the [Formula: see text]-tangle [Formula: see text] in a way that preserves the orientation of [Formula: see text]. After developing a general decomposition for the Jones polynomial of the virtual link [Formula: see text] in terms of various (modified) closures of [Formula: see text], we analyze the Jones polynomials of virtual links [Formula: see text] that differ via a local move of type [Formula: see text]. Succinct divisibility conditions on [Formula: see text] are derived for broad classes of local moves that include the [Formula: see text]-move and the double-[Formula: see text]-move as special cases. As a consequence of our divisibility result for the double-[Formula: see text]-move, we introduce a necessary condition for any pair of classical knots to be [Formula: see text]-equivalent.

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ON THE JONES POLYNOMIAL OF ADEQUATE VIRTUAL LINKS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216510008212 ◽

2010 ◽

Vol 19 (07) ◽

pp. 961-974

Author(s):

YONGJU BAE ◽

HYE SOOK LEE ◽

CHAN-YOUNG PARK

Keyword(s):

Jones Polynomial ◽

Crossing Number ◽

Virtual Link ◽

Link Diagram ◽

Virtual Links

In this paper, we prove that an adequate virtual link diagram of an adequate virtual link has minimal real crossing number.

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SEQUAL: an interactive computer program for sequential classification of biomedical data

Computer Methods and Programs in Biomedicine ◽

10.1016/0169-2607(88)90085-5 ◽

1988 ◽

Vol 27 (3) ◽

pp. 213-221

Author(s):

B. Bona ◽

R. Tempo ◽

G. Belforte

Keyword(s):

Computer Program ◽

Biomedical Data ◽

Interactive Computer Program ◽

Sequential Classification

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A MULTI-VARIABLE POLYNOMIAL INVARIANT FOR UNORIENTED VIRTUAL KNOTS AND LINKS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216509007099 ◽

2009 ◽

Vol 18 (05) ◽

pp. 625-649 ◽

Cited By ~ 5

Author(s):

YASUYUKI MIYAZAWA

Keyword(s):

Crossing Number ◽

Weighted Sum ◽

Virtual Link ◽

Polynomial Invariant ◽

Virtual Knots ◽

Virtual Links ◽

Knots And Links

We construct a multi-variable polynomial invariant Y for unoriented virtual links as a certain weighted sum of polynomials, which are derived from virtual magnetic graphs with oriented vertices, on oriented virtual links associated with a given virtual link. We show some features of the Y-polynomial including an evaluation of the virtual crossing number of a virtual link.

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MHPT.BAS: a computer program for modified Hill–Piper diagram for classification of ground water

Computers & Geosciences ◽

10.1016/s0098-3004(98)00083-1 ◽

1998 ◽

Vol 24 (10) ◽

pp. 991-1008 ◽

Cited By ~ 15

Author(s):

N.Srinivasa Rao

Keyword(s):

Computer Program ◽

Ground Water ◽

Piper Diagram

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SOME NUMERICAL INVARIANTS OF FLAT VIRTUAL LINKS ARE ALWAYS REALIZED BY REDUCED DIAGRAMS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216506004476 ◽

2006 ◽

Vol 15 (03) ◽

pp. 289-297 ◽

Cited By ~ 4

Author(s):

TERUHISA KADOKAMI

Keyword(s):

Intersection Number ◽

Finite Sequence ◽

Closed Surface ◽

Crossing Number ◽

The Self ◽

Virtual Link ◽

Virtual Knot ◽

Virtual Links ◽

Geodesic Loop ◽

Numerical Invariants

Any flat virtual link has a reduced diagram which satisfies a certain minimality, and reduced diagrams are related one another by a finite sequence of a certain Reidemeister move. The move preserves some numerical invariants of diagrams. So we can define numerical invariants for flat virtual links. One of them, the crossing number of a flat virtual knot K, coinsides with the self-intersection number of K as an essential geodesic loop on a hyperbolic closed surface. We also show an equation among these numerical invariants, basic properties by using the equation, and determine non-split flat virtual links with the crossing number up to three.

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Graphs, Links, and Duality on Surfaces

Combinatorics Probability Computing ◽

10.1017/s0963548310000295 ◽

2010 ◽

Vol 20 (2) ◽

pp. 267-287 ◽

Cited By ~ 14

Author(s):

VYACHESLAV KRUSHKAL

Keyword(s):

Planar Graphs ◽

Jones Polynomial ◽

Tutte Polynomial ◽

Natural Duality ◽

Polynomial Invariant ◽

Topological Duality ◽

Ribbon Graphs ◽

Virtual Links ◽

Graphs On Surfaces ◽

Tutte Polynomials

We introduce a polynomial invariant of graphs on surfaces,PG, generalizing the classical Tutte polynomial. Topological duality on surfaces gives rise to a natural duality result forPG, analogous to the duality for the Tutte polynomial of planar graphs. This property is important from the perspective of statistical mechanics, where the Tutte polynomial is known as the partition function of the Potts model. For ribbon graphs,PGspecializes to the well-known Bollobás–Riordan polynomial, and in fact the two polynomials carry equivalent information in this context. Duality is also established for a multivariate version of the polynomialPG. We then consider a 2-variable version of the Jones polynomial for links in thickened surfaces, taking into account homological information on the surface. An analogue of Thistlethwaite's theorem is established for these generalized Jones and Tutte polynomials for virtual links.

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Cyclic coverings of virtual link diagrams

International Journal of Mathematics ◽

10.1142/s0129167x19500721 ◽

2019 ◽

Vol 30 (14) ◽

pp. 1950072 ◽

Cited By ~ 1

Author(s):

Naoko Kamada

Keyword(s):

Virtual Link ◽

Link Diagram ◽

Cyclic Covering ◽

Link Diagrams ◽

Virtual Links ◽

Cyclic Coverings

A virtual link diagram is called mod [Formula: see text] almost classical if it admits an Alexander numbering valued in integers modulo [Formula: see text], and a virtual link is called mod [Formula: see text] almost classical if it has a mod [Formula: see text] almost classical diagram as a representative. In this paper, we introduce a method of constructing a mod [Formula: see text] almost classical virtual link diagram from a given virtual link diagram, which we call an [Formula: see text]-fold cyclic covering diagram. The main result is that [Formula: see text]-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 [Formula: see text] almost classical virtual links. Some applications are also given.

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The generalized Alexander polynomial of periodic virtual links

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216519500421 ◽

2019 ◽

Vol 28 (06) ◽

pp. 1950042

Author(s):

Joonoh Kim ◽

Kyoung-Tark Kim ◽

Mi Hwa Shin

Keyword(s):

Alexander Polynomial ◽

State Model ◽

Quantum Algebras ◽

Virtual Link ◽

Linking Numbers ◽

Virtual Links ◽

Polynomial Formula

In this paper, we give several simple criteria to detect possible periods and linking numbers for a given virtual link. We investigate the behavior of the generalized Alexander polynomial [Formula: see text] of a periodic virtual link [Formula: see text] via its Yang–Baxter state model given in [L. H. Kauffman and D. E. Radford, Bi-oriented quantum algebras and a generalized Alexander polynomial for virtual links, in Diagrammatic Morphisms and Applications, Contemp. Math. 318 (2003) 113–140, arXiv:math/0112280v2 [math.GT] 31 Dec 2001].

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