scholarly journals Introduction to disoriented knot theory

2018 ◽  
Vol 16 (1) ◽  
pp. 346-357
Author(s):  
İsmet Altıntaş
Keyword(s):  
Knot Theory ◽  
Jones Polynomial ◽  
Polynomial Invariants ◽  
Linking Number ◽  
Link Diagrams ◽  
New Ideas ◽  
Numerical Invariants ◽  
Bracket Polynomial ◽  
Knots And Links

AbstractThis paper is an introduction to disoriented knot theory, which is a generalization of the oriented knot and link diagrams and an exposition of new ideas and constructions, including the basic definitions and concepts such as disoriented knot, disoriented crossing and Reidemesiter moves for disoriented diagrams, numerical invariants such as the linking number and the complete writhe, the polynomial invariants such as the bracket polynomial, the Jones polynomial for the disoriented knots and links.

scholarly journals A POLYNOMIAL INVARIANT OF VIRTUAL LINKS

2013 ◽  
Vol 22 (12) ◽  
pp. 1341002 ◽  
Author(s):  
ZHIYUN CHENG ◽  
HONGZHU GAO
Keyword(s):  
Knot Theory ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knots ◽  
The Third ◽  
Linking Number ◽  
Virtual Links ◽  
Definition Of ◽  
Knots And Links

In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms [Parity in knot theory, Sb. Math.201 (2010) 693–733] to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author in [A polynomial invariant of virtual knots, preprint (2012), arXiv:math.GT/1202.3850v1]. The relation between this new polynomial invariant and the affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramification22 (2013) 1340007; A linking number definition of the affine index polynomial and applications, preprint (2012), arXiv:1211.1747v1] is discussed. In the second part we introduce a polynomial invariant for long flat virtual knots. In the third part we define a polynomial invariant for 2-component virtual links. This polynomial invariant can be regarded as a generalization of the linking number.

ON A LOCAL MOVE FOR VIRTUAL KNOTS AND LINKS

2009 ◽  
Vol 18 (11) ◽  
pp. 1577-1596 ◽  
Author(s):  
TOSHIYUKI OIKAWA
Keyword(s):  
Sufficient Conditions ◽  
Virtual Link ◽  
Link Invariant ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Necessary And Sufficient ◽  
Knots And Links

We define a local move called a CF-move on virtual link diagrams, and show that any virtual knot can be deformed into a trivial knot by using generalized Reidemeister moves and CF-moves. Moreover, we define a new virtual link invariant n(L) for a virtual 2-component link L whose virtual linking number is an integer. Then we give necessary and sufficient conditions for two virtual 2-component links to be deformed into each other by using generalized Reidemeister moves and CF-moves in terms of a virtual linking number and n(L).

Knots, matroids and the Ising model

1993 ◽  
Vol 113 (1) ◽  
pp. 107-139 ◽  
Author(s):  
W. Schwärzler ◽  
D. J. A. Welsh
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Knot Theory ◽  
Tutte Polynomial ◽  
Signed Graphs ◽  
Kauffman Bracket ◽  
Link Diagrams ◽  
Bracket Polynomial

AbstractA polynomial is defined on signed matroids which contains as specializations the Kauffman bracket polynomial of knot theory, the Tutte polynomial of a matroid, the partition function of the anisotropic Ising model, the Kauffman–Murasugi polynomials of signed graphs. It leads to generalizations of a theorem of Lickorish and Thistlethwaite showing that adequate link diagrams do not represent the unknot. We also investigate semi-adequacy and the span of the bracket polynomial in this wider context.

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

2017 ◽  
Vol 26 (03) ◽  
pp. 1741001 ◽  
Author(s):  
Heather A. Dye ◽  
Aaron Kaestner ◽  
Louis H. Kauffman
Keyword(s):  
Large Class ◽  
Jones Polynomial ◽  
Khovanov Homology ◽  
Frobenius Algebra ◽  
Conjugation Operator ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Link Diagrams ◽  
Knots And Links

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 knots 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 is non-classical, 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 [Formula: see text] in the sense of Turaev [42].

Parities and polynomial invariants for virtual links

2014 ◽  
Vol 23 (12) ◽  
pp. 1450066 ◽  
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.

scholarly journals Knot theory realizations in nematic colloids

2015 ◽  
Vol 112 (6) ◽  
pp. 1675-1680 ◽  
Author(s):  
Simon Čopar ◽  
Uroš Tkalec ◽  
Igor Muševič ◽  
Slobodan Žumer
Keyword(s):  
Phase Diagram ◽  
Knot Theory ◽  
Colloidal Particles ◽  
Jones Polynomial ◽  
Laser Tweezers ◽  
Knot Invariants ◽  
Topological Parameters ◽  
Twisted Nematic ◽  
2D Arrays ◽  
Knots And Links

Nematic braids are reconfigurable knots and links formed by the disclination loops that entangle colloidal particles dispersed in a nematic liquid crystal. We focus on entangled nematic disclinations in thin twisted nematic layers stabilized by 2D arrays of colloidal particles that can be controlled with laser tweezers. We take the experimentally assembled structures and demonstrate the correspondence of the knot invariants, constructed graphs, and surfaces associated with the disclination loop to the physically observable features specific to the geometry at hand. The nematic nature of the medium adds additional topological parameters to the conventional results of knot theory, which couple with the knot topology and introduce order into the phase diagram of possible structures. The crystalline order allows the simplified construction of the Jones polynomial and medial graphs, and the steps in the construction algorithm are mirrored in the physics of liquid crystals.

scholarly journals INTRODUCTION TO VIRTUAL KNOT THEORY

2012 ◽  
Vol 21 (13) ◽  
pp. 1240007 ◽  
Author(s):  
LOUIS H. KAUFFMAN
Keyword(s):  
Knot Theory ◽  
Virtual Knot ◽  
New Ideas ◽  
Bracket Polynomial ◽  
Virtual Knot Theory

This paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial and categorifications of the arrow polynomial.

Relative Tutte Polynomials for Coloured Graphs and Virtual Knot Theory

2009 ◽  
Vol 19 (3) ◽  
pp. 343-369 ◽  
Author(s):  
Y. DIAO ◽  
G. HETYEI
Keyword(s):  
Knot Theory ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Ribbon Graph ◽  
Virtual Knot ◽  
Tree Expansion ◽  
The Face ◽  
Tutte Polynomials ◽  
Bracket Polynomial ◽  
Virtual Knot Theory

We introduce the concept of a relative Tutte polynomial of coloured graphs. We show that this relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this subject. We then apply the relative Tutte polynomial to virtual knot theory. More specifically, we show that the Kauffman bracket polynomial (and hence the Jones polynomial) of a virtual knot can be computed from the relative Tutte polynomial of its face (Tait) graph with some suitable variable substitutions. Our method offers an alternative to the ribbon graph approach, using the face graph obtained from the virtual link diagram directly.

scholarly journals Alexander and writhe polynomials for virtual knots

2016 ◽  
Vol 25 (08) ◽  
pp. 1650050 ◽  
Author(s):  
Blake Mellor
Keyword(s):  
Knot Theory ◽  
Alexander Polynomial ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Virtual Links ◽  
New Interpretation ◽  
Polynomial Formula ◽  
Knots And Links

We give a new interpretation of the Alexander polynomial [Formula: see text] for virtual knots due to Sawollek [On Alexander–Conway polynomials for virtual knots and Links, preprint (2001), arXiv:math/9912173] and Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications 12 (2003) 987–1000], and use it to show that, for any virtual knot, [Formula: see text] determines the writhe polynomial of Cheng and Gao [A polynomial invariant of virtual links, J. Knot Theory Ramifications 22(12) (2013), Article ID: 1341002, 33pp.] (equivalently, Kauffman’s affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramifications 22(4) (2013), Article ID: 1340007, 30pp.]). We also use it to define a second-order writhe polynomial, and give some applications.

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