scholarly journals Polynomials and Homotopy of Virtual Knot Diagrams

2017 ◽  
Vol 57 (1) ◽  
pp. 145-161
Author(s):  
Myeong-Ju Jeong ◽  
Chan-Young Park ◽  
Maeng Sang Park
Keyword(s):  
Virtual Knot ◽  
Knot Diagrams

Picture-valued parity-biquandle bracket

2020 ◽  
Vol 29 (02) ◽  
pp. 2040004 ◽  
Author(s):  
Denis P. Ilyutko ◽  
Vassily O. Manturov
Keyword(s):  
Classical Case ◽  
Linear Combinations ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Underlying Graph ◽  
Kauffman Bracket ◽  
Knot Diagrams

In V. O. Manturov, On free knots, preprint (2009), arXiv:math.GT/0901.2214], the second named author constructed the bracket invariant [Formula: see text] of virtual knots valued in pictures (linear combinations of virtual knot diagrams with some crossing information omitted), such that for many diagrams [Formula: see text], the following formula holds: [Formula: see text], where [Formula: see text] is the underlying graph of the diagram, i.e. the value of the invariant on a diagram equals the diagram itself with some crossing information omitted. This phenomenon allows one to reduce many questions about virtual knots to questions about their diagrams. In [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, preprint (2015), arXiv:math.GT/1508.06573], the authors discovered the following phenomenon: having a biquandle coloring of a certain knot, one can enhance various state-sum invariants (say, Kauffman bracket) by using various coefficients depending on colors. Taking into account that the parity can be treated in terms of biquandles, we bring together the two ideas from these papers and construct the picture-valued parity-biquandle bracket for classical and virtual knots. This is an invariant of virtual knots valued in pictures. Both the parity bracket and Nelson–Orrison–Rivera invariants are partial cases of this invariant, hence this invariant enjoys many properties of various kinds. Recently, the authors together with E. Horvat and S. Kim have found that the picture-valued phenomenon works in the classical case.

Reidemeister moves and parity polynomials of virtual knot diagrams

2017 ◽  
Vol 26 (10) ◽  
pp. 1750051
Author(s):  
Myeong-Ju Jeong
Keyword(s):  
Lower Bounds ◽  
Virtual Knot ◽  
The Third ◽  
Reidemeister Moves ◽  
Polynomial Formula ◽  
Knot Diagrams

When two virtual knot diagrams are virtually isotopic, there is a sequence of Reidemeister moves and virtual moves relating them. I introduced a polynomial [Formula: see text] of a virtual knot diagram [Formula: see text] and gave lower bounds for the number of Reidemeister moves in deformation of virtually isotopic knot diagrams by using [Formula: see text]. In this paper, I introduce bridge diagrams and polynomials of virtual knot diagrams based on parity of crossings, and show that the polynomials give lower bounds for the number of the third Reidemeister moves. I give an example which shows that the result is distinguished from that obtained from [Formula: see text].

Reidemeister moves and a polynomial of virtual knot diagrams

2015 ◽  
Vol 24 (02) ◽  
pp. 1550010 ◽  
Author(s):  
Myeong-Ju Jeong
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Virtual Knot ◽  
Reidemeister Moves ◽  
Knot Diagrams

In 2006 C. Hayashi gave a lower bound for the number of Reidemeister moves in deformation of two equivalent knot diagrams by using writhe and cowrithe. It can be naturally extended for two virtually isotopic virtual knot diagrams. We introduce a polynomial qK(t) of a virtual knot diagram K and give lower bounds for the number of Reidemeister moves in deformation of two virtually isotopic knots by using qK(t). We give an example which shows that the polynomial qK(t) is useful to map out a sequence of Reidemeister moves to deform a virtual knot diagram to another virtually isotopic one.

Geodesic graphs for a homotopy class of virtual knots

2017 ◽  
Vol 26 (13) ◽  
pp. 1750090
Author(s):  
Sumiko Horiuchi ◽  
Yoshiyuki Ohyama
Keyword(s):  
Homotopy Class ◽  
Finite Sequence ◽  
Dimensional Lattice ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Minimum Number ◽  
Vertex Set ◽  
Number Formula ◽  
Lattice Graph ◽  
Knot Diagrams

We consider a local move, denoted by [Formula: see text], on knot diagrams or virtual knot diagrams.If two (virtual) knots [Formula: see text] and [Formula: see text] are transformed into each other by a finite sequence of [Formula: see text] moves, the [Formula: see text] distance between [Formula: see text] and [Formula: see text] is the minimum number of times of [Formula: see text] moves needed to transform [Formula: see text] into [Formula: see text]. By [Formula: see text], we denote the set of all (virtual) knots which can be transformed into a (virtual) knot [Formula: see text] by [Formula: see text] moves. A geodesic graph for [Formula: see text] is the graph which satisfies the following: The vertex set consists of (virtual) knots in [Formula: see text] and for any two vertices [Formula: see text] and [Formula: see text], the distance on the graph from [Formula: see text] to [Formula: see text] coincides with the [Formula: see text] distance between [Formula: see text] and [Formula: see text]. When we consider virtual knots and a crossing change as a local move [Formula: see text], we show that the [Formula: see text]-dimensional lattice graph for any given natural number [Formula: see text] and any tree are geodesic graphs for [Formula: see text].

A PARITY AND A MULTI-VARIABLE POLYNOMIAL INVARIANT FOR VIRTUAL LINKS

2013 ◽  
Vol 22 (13) ◽  
pp. 1350073 ◽  
Author(s):  
YOUNG HO IM ◽  
KYOUNG IL PARK
Keyword(s):  
Knot Theory ◽  
Virtual Link ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Link Diagrams ◽  
Virtual Links ◽  
Knot Diagrams

We introduce a parity of classical crossings of virtual link diagrams which extends the Gaussian parity of virtual knot diagrams and the odd writhe of virtual links that extends that of virtual knots introduced by Kauffman [A self-linking invariants of virtual knots, Fund. Math.184 (2004) 135–158]. Also, we introduce a multi-variable polynomial invariant for virtual links by using the parity of classical crossings, which refines the index polynomial introduced in [Index polynomial invariants of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. As consequences, we give some properties of our invariant, and raise some examples.

scholarly journals FILAMENTATIONS FOR VIRTUAL LINKS

2006 ◽  
Vol 15 (03) ◽  
pp. 327-338 ◽  
Author(s):  
WILLIAM J. SCHELLHORN
Keyword(s):  
Virtual Knot ◽  
Virtual Knots ◽  
Chord Diagrams ◽  
Virtual Links ◽  
Knot Diagrams

In 2002, Hrencecin and Kauffman defined a filamentation invariant on oriented chord diagrams that may determine whether the corresponding flat virtual knot diagrams are non-trivial. A virtual knot diagram is non-classical if its related flat virtual knot diagram is non-trivial. Hence filamentations can be used to detect non-classical virtual knots. We extend these filamentation techniques to virtual links with more than one component. We also give examples of virtual links that they can detect as non-classical.

scholarly journals VIRTUAL KNOT DIAGRAMS AND THE WITTEN–RESHETIKHIN–TURAEV INVARIANT

2005 ◽  
Vol 14 (08) ◽  
pp. 1045-1075 ◽  
Author(s):  
H. A. DYE ◽  
LOUIS H. KAUFFMAN
Keyword(s):  
Fundamental Group ◽  
Virtual Link ◽  
Virtual Knot ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Classical Link ◽  
Kirby Calculus ◽  
Knot Diagrams

The Witten–Reshetikhin–Turaev invariant of classical link diagrams is generalized to virtual link diagrams. This invariant is unchanged by the framed Reidemeister moves and the Kirby calculus. As a result, it is also an invariant of the 3-manifolds represented by the classical link diagrams. This generalization is used to demonstrate that there are virtual knot diagrams with a non-trivial Witten–Reshetikhin–Turaev invariant and trivial 3-manifold fundamental group.

Classification of genus 1 virtual knots having at most five classical crossings

2014 ◽  
Vol 23 (06) ◽  
pp. 1450031 ◽  
Author(s):  
Akimova Alena Andreevna ◽  
Sergei Vladimirovich Matveev
Keyword(s):  
Virtual Knot ◽  
Virtual Knots ◽  
Kauffman Polynomial ◽  
Thickened Torus ◽  
Knot Diagrams ◽  
Genus One

The goal of this paper is to tabulate all genus one prime virtual knots having diagrams with ≤ 5 classical crossings. First, we construct all nonlocal prime knots in the thickened torus T × I which have diagrams with ≤ 5 crossings and admit no destabilizations. Then we use a generalized version of the Kauffman polynomial to prove that all those knots are different. Finally, we convert the knot diagrams in T thus obtained into virtual knot diagrams in the plane.

scholarly journals Kauffman–Harary conjecture for alternating virtual knots

2015 ◽  
Vol 24 (08) ◽  
pp. 1550046
Author(s):  
Zhiyun Cheng
Keyword(s):  
Affirmative Answer ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Prime Determinant ◽  
Knot Diagrams

In 1999, Kauffman–Harary conjectured that every non-trivial Fox p-coloring of a reduced, alternating knot diagram with prime determinant p is heterogeneous. Ten years later this conjecture was proved by Mattman and Solis. Williamson generalized this conjecture to alternating virtual knots and proved it for certain families of alternating virtual knots. In this paper, by studying the coloring matrices and the determinants of alternating virtual knot diagrams we give an affirmative answer to the Kauffman–Harary conjecture for alternating virtual knots.

Characterization of signed Gauss paragraphs and skew-symmetric graded matrices

2018 ◽  
Vol 27 (01) ◽  
pp. 1850002 ◽  
Author(s):  
José Gregorio Rodríguez-Nieto
Keyword(s):  
Embedded Graphs ◽  
Intersection Pairing ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Virtual Links ◽  
The One ◽  
Minimal Realizations ◽  
Knot Diagrams ◽  
Bracket Polynomials

In this paper, we use theory of embedded graphs on oriented and compact [Formula: see text]-surfaces to construct minimal realizations of signed Gauss paragraphs. We prove that the genus of the ambient surface of these minimal realizations can be seen as a function of the maximum number of Carter’s circles. For the case of signed Gauss words, we use a generating set of [Formula: see text], given in [G. Cairns and D. Elton, The Planarity problem for signed Gauss world, J. Knots Theor. Ramif. 2(4) (1993) 359–367], and the intersection pairing of immersed [Formula: see text]-normal curves to present a short solution of the signed Gauss word problem. We relate this solution with the one given by Cairns and Elton. Moreover, we define the join operation on signed Gauss paragraphs to produce signed Gauss words such that both can be realized on the same minimal genus [Formula: see text]-surface. We connect the characterization of signed Gauss paragraph with the recognition virtual links problem. Also we present a combinatorial algorithm to compute, in an easier way, skew-symmetric graded matrices [V. Turaev, Cobordism of knots on surfaces, J. Topol. 1(2) (2008) 285–305] for virtual knots through the concept of triplets [M. Toro and J. Rodríguez, Triplets associated to virtual knot diagrams, Rev. Integración (2011)]. Therefore, we can prove that the Kishino’s knot is not classical, moreover, we prove that the virtual knots of the family [Formula: see text] given in [H. A. Dye, Virtual knots undetected by [Formula: see text] and [Formula: see text]-strand bracket polynomials, Topol. Appl. 153 (2005) 141–160] are not classical knots.

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