A polynomial invariant of virtual knots

Cheng, Zhongtao

doi:10.1090/s0002-9939-2013-11785-5

Cited by 51 publications

(50 citation statements)

References 7 publications

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“…The Odd Wriggle Polynomial can be shown to be equivalent to the Odd Writhe Polynomial defined by Cheng [2]. Equality occurs after multiplying the Odd Writhe Polynomial by t −1 .…”

Section: Theorem 3 the Wriggle Polynomial W K (T) Is A Virtual Knomentioning

confidence: 99%

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A Linking Number Definition of the Affine Index Polynomial and Applications

Folwaczny

Kauffman

2013

J. Knot Theory Ramifications

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This paper gives an alternate definition of the Affine Index Polynomial (called the Wriggle Polynomial) using virtual linking numbers and explores applications of this polynomial. In particular, it proves the Cosmetic Crossing Change Conjecture for odd virtual knots and pure virtual knots. It also demonstrates that the polynomial can detect mutations by positive rotation and proves it cannot detect mutations by positive reflection. Finally it exhibits a pair of mutant knots that can be distinguished by a type 2 vassiliev invariant coming from the polynomial.

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“…The Odd Wriggle Polynomial can be shown to be equivalent to the Odd Writhe Polynomial defined by Cheng [2]. Equality occurs after multiplying the Odd Writhe Polynomial by t −1 .…”

Section: Theorem 3 the Wriggle Polynomial W K (T) Is A Virtual Knomentioning

confidence: 99%

“…The labeling system defined in part (2) of the lemma was first defined by Cheng [2] and a different weight was calculated. As a consequence of this lemma, we may assume that each arc in the Gauss diagram is labeled by the sum of signs of chords first encountered as over (i.e.…”

Section: Lemma 2 (Kauffman [14])mentioning

confidence: 99%

A Linking Number Definition of the Affine Index Polynomial and Applications

Folwaczny

Kauffman

2013

J. Knot Theory Ramifications

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“…(3) The concatenation of +S ij (n) and −S ij (n) is S-equivalent to that of n copies of +S i (1) for n > 0 or n copies of canceling pairs for n < 0 by using Lemma 2.3 (2) and Corollary 2.4 (1). See Figure 13, where ε is the sign of n; that is, n = ε|n|.…”

Section: Figure 2 Self-and Nonself-chords Of a Gauss Diagrammentioning

confidence: 99%

Writhe polynomials and shell moves for virtual knots and links

Nakamura

Nakanishi

Satoh

2020

European Journal of Combinatorics

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The writhe polynomial is a fundamental invariant of an oriented virtual knot. We introduce a kind of local moves for oriented virtual knots called shell moves. The first aim of this paper is to prove that two oriented virtual knots have the same writhe polynomial if and only if they are related by a finite sequence of shell moves. The second aim of this paper is to classify oriented 2-component virtual links up to shell moves by using several invariants of virtual links.

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“…Virtual knots were introduced by Kauffman [11] as a generalization of classical knot theory, and since then many invariants have been developed to help distinguish virtual knots, and to determine when a virtual knot is equivalent to a classical knot. In the past few years, several authors have developed invariants that generalize the classical writhe of a knot [3,4,6,9,12,13,17]. These invariants have been used to define Vassiliev invariants of virtual knots [9,13], give bounds on the unknotting number (when it exists) and forbidden number of virtual knots [17,5], and distinguish mutant virtual knots [6], among other applications.…”

Section: Introductionmentioning

confidence: 99%