Oriented local moves and divisibility of the Jones–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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Clasp-pass moves and Kauffman polynomials of virtual knots

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216516500450 ◽

2016 ◽

Vol 25 (08) ◽

pp. 1650045

Author(s):

Myeong-Ju Jeong ◽

Dahn-Goon Kim

Keyword(s):

Lower Bound ◽

Jones Polynomial ◽

Finite Sequence ◽

Vassiliev Invariants ◽

Type Formula ◽

Virtual Knots ◽

Vassiliev Invariant ◽

Kauffman Polynomial

Habiro showed that two knots [Formula: see text] and [Formula: see text] are related by a finite sequence of clasp-pass moves, if and only if they have the same value for Vassiliev invariants of type [Formula: see text]. Tsukamoto showed that, if two knots differ by a clasp-pass move then the values of the Vassiliev invariant [Formula: see text] of degree [Formula: see text] for the two knots differ by [Formula: see text] or [Formula: see text], where [Formula: see text] is the Jones polynomial of a knot [Formula: see text]. If two virtual knots are related by clasp-pass moves, then they take the same value for all Vassiliev invariants of degree [Formula: see text]. We extend the Tsukamoto’s result to virtual knots by using a Vassiliev invariant [Formula: see text] of degree [Formula: see text], which is induced from the Kauffman polynomial. We also get a lower bound for the minimal number of clasp-pass moves needed to transform [Formula: see text] to [Formula: see text], if two virtual knots [Formula: see text] and [Formula: see text] can be related by a finite sequence of clasp-pass moves.

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Virtual Khovanov homology using cobordisms

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216514500461 ◽

2014 ◽

Vol 23 (09) ◽

pp. 1450046 ◽

Cited By ~ 2

Author(s):

Daniel Tubbenhauer

Keyword(s):

Computer Program ◽

Jones Polynomial ◽

Virtual Link ◽

Link Homology ◽

Direct Extension ◽

Virtual Links ◽

Local Properties ◽

Characteristic Two ◽

Topological Construction

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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Parities and polynomial invariants for virtual links

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216514500667 ◽

2014 ◽

Vol 23 (12) ◽

pp. 1450066 ◽

Cited By ~ 3

Author(s):

Young Ho Im ◽

Kyoung Il Park ◽

Mi Hwa Shin

Keyword(s):

Knot Theory ◽

Virtual Link ◽

Polynomial Invariant ◽

Polynomial Invariants ◽

Virtual Knots ◽

Link Diagrams ◽

Virtual Links ◽

Kauffman Polynomial

We introduce the odd Jones–Kauffman polynomial and odd Miyazawa polynomials of virtual link diagrams by using the parity of virtual link diagrams given in [Y. H. Im and K. I. Park, A parity and a multi-variable polynomial invariant for virtual links, J. Knot Theory Ramifications22(13) (2013), Article ID: 1350073, 18pp.], which are different from the original Jones–Kauffman and Miyazawa polynomials. Also, we give a family of parities and odd polynomials for virtual knots so that many virtual knots can be distinguished.

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State sum invariants for flat virtual links from the chord index

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216520500273 ◽

2020 ◽

Vol 29 (05) ◽

pp. 2050027

Author(s):

Kyeonghui Lee ◽

Young Ho Im ◽

Sera Kim

Keyword(s):

Virtual Link ◽

Polynomial Invariants ◽

Link Diagrams ◽

Virtual Links ◽

Kauffman Polynomial

We introduce some polynomial invariants for flat virtual links which are similar to the Jones–Kauffman polynomial, the Miyazawa polynomial and the arrow polynomial for virtual link diagrams, and we give several properties and examples.

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MAGNETIC GRAPHS AND AN INVARIANT FOR VIRTUAL LINKS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216506005135 ◽

2006 ◽

Vol 15 (10) ◽

pp. 1319-1334 ◽

Cited By ~ 30

Author(s):

YASUYUKI MIYAZAWA

Keyword(s):

Virtual Link ◽

Polynomial Invariant ◽

Link Invariant ◽

Virtual Links ◽

Kauffman Polynomial

Introducing a type of graph named a virtual magnetic graph diagram, we define a virtual link invariant, which generalizes the Jones-Kauffman polynomial and the 2-variable polynomial invariant defined by N. Kamada and the author. We show that this invariant can evaluate the number of virtual crossings of a virtual link.

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Span of the Jones polynomials of certain v-adequate virtual links

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216521500012 ◽

2021 ◽

pp. 2150001

Author(s):

Minori Okamura ◽

Keiichi Sakai

Keyword(s):

Type Inequality ◽

Virtual Link ◽

Link Diagram ◽

Virtual Links ◽

Jones Polynomials

It is known that the Kauffman–Murasugi–Thislethwaite type inequality becomes an equality for any (possibly virtual) adequate link diagram. We refine this condition. As an application we obtain a criterion for virtual link diagram with exactly one virtual crossing to represent a properly virtual link.

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Similarity indices and the Miyazawa polynomials of virtual links

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216514600037 ◽

2014 ◽

Vol 23 (07) ◽

pp. 1460003 ◽

Cited By ~ 1

Author(s):

Myeong-Ju Jeong ◽

Chan-Young Park

Keyword(s):

Knot Theory ◽

State Model ◽

Necessary Condition ◽

Virtual Link ◽

Polynomial Invariant ◽

Virtual Knots ◽

Vassiliev Invariant ◽

Virtual Links ◽

Bracket Polynomial ◽

Knots And Links

Y. Miyazawa introduced a two-variable polynomial invariant of virtual knots in 2006 [Magnetic graphs and an invariant for virtual links, J. Knot Theory Ramifications 15 (2006) 1319–1334] and then generalized it to give a multi-variable one via decorated virtual magnetic graph diagrams in 2008. A. Ishii gave a simple state model for the two-variable Miyazawa polynomial by using pole diagrams in 2008 [A multi-variable polynomial invariant for virtual knots and links, J. Knot Theory Ramifications 17 (2008) 1311–1326]. H. A. Dye and L. H. Kauffman constructed an arrow polynomial of a virtual link in 2009 which is equivalent to the multi-variable Miyazawa polynomial [Virtual crossing number and the arrow polynomial, preprint (2008), arXiv:0810.3858v3, http://front.math.ucdavis.edu .]. We give a bracket model for the multi-variable Miyazawa polynomial via pole diagrams and polar tangles similarly to the Ishii's state model for the two-variable polynomial. By normalizing the bracket polynomial we get the multi-variable Miyazawa polynomial fK ∈ ℤ[A, A-1, K1, K2, …] of a virtual link K. n-similar knots take the same value for any Vassiliev invariant of degree < n. We show that fK1 ≡ fK2 mod (A4 - 1)n if two virtual links K1 and K2 are n-similar. Also we give a necessary condition for a virtual link to be periodic by using n-similarity of virtual tangles and the Miyazawa polynomial.

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MULTI-VARIABLE POLYNOMIAL INVARIANTS FOR VIRTUAL LINKS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216503002962 ◽

2003 ◽

Vol 12 (08) ◽

pp. 1131-1144 ◽

Cited By ~ 11

Author(s):

VASSILY O. MANTUROV

Keyword(s):

Virtual Link ◽

Invariant Polynomials ◽

Polynomial Invariants ◽

Link Invariants ◽

Virtual Knots ◽

Virtual Links ◽

Multiple Variables ◽

Kauffman Polynomial ◽

Knots And Links

We construct new invariant polynomials in two and multiple variables for virtual knots and links. They are defined as determinants of Alexander-like matrices whose determinants are virtual link invariants. These polynomials vanish on classical links. In some cases, they separate links that can not be separated by the Jones–Kauffman polynomial [Kau] and the polynomial proposed in [Ma3].

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THE VIRTUAL CROSSING NUMBER OF 2-COMPONENT FLAT VIRTUAL LINKS WITH THE SUPPORTING GENUS ONE

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216508006300 ◽

2008 ◽

Vol 17 (05) ◽

pp. 633-647

Author(s):

TERUHISA KADOKAMI

Keyword(s):

Crossing Number ◽

Virtual Link ◽

Virtual Links ◽

Special Cases ◽

Genus One ◽

Surface Realization

We introduce a method to estimate the virtual crossing number of a 2-component flat virtual link. In particular, we give an estimation for the case that a surface realization of the link is expressed by two non-homologous simple loops on the torus (whose supporting genus is one automatically), and determine for some special cases. To know estimations, it is important to consider "realizing problem".

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