scholarly journals Band surgery on knots and links, III

2016 ◽  
Vol 25 (10) ◽  
pp. 1650056 ◽  
Author(s):  
Taizo Kanenobu
Keyword(s):  
Jones Polynomial ◽  
Special Value ◽  
Knots And Links

We give two criteria of links concerning a band surgery: The first one is a condition on the determinants of links which are related by a band surgery using Nakanishi’s criterion on knots with Gordian distance one. The second one is a criterion on knots with [Formula: see text]-Gordian distance two by using a special value of the Jones polynomial, where an [Formula: see text]-move is a band surgery preserving a component number. Then, we give an improved table of [Formula: see text]-Gordian distances between knots with up to seven crossings, where we add Zeković’s result.

scholarly journals OCNEANU'S TRACE AND STARKEY'S RULE

2003 ◽  
Vol 12 (07) ◽  
pp. 899-904 ◽  
Author(s):  
MEINOLF GECK ◽  
NICOLAS JACON
Keyword(s):  
Simple Proof ◽  
Hecke Algebras ◽  
Jones Polynomial ◽  
Main Tool ◽  
Type A ◽  
Character Tables ◽  
Special Case ◽  
Knots And Links

We give a new simple proof for the weights of Ocneanu's trace on Iwahori–Hecke algebras of type A. This trace is used in the construction of the HOMFLYPT-polynomial of knots and links (which includes the famous Jones polynomial as a special case). Our main tool is Starkey's rule concerning the character tables of Iwahori–Hecke algebras of type A.

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 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].

scholarly journals Oriented Gordian distance of two-component links with up to seven crossings

2015 ◽  
Vol 24 (10) ◽  
pp. 1540013 ◽  
Author(s):  
Saori Kanenobu ◽  
Taizo Kanenobu
Keyword(s):  
Jones Polynomial ◽  
The Other ◽  
Two Component ◽  
Special Value

The oriented Gordian distance between two oriented links is the minimal number of crossing changes needed to deform one into the other. We compile a table of oriented Gordian distances between 2-component non-splittable links with up to six crossings. In particular, we give a criterion of oriented Gordian distance two using a special value of the Jones polynomial, which allows us to prove that the unlinking number of the 2-component link [Formula: see text] is 3. This is one of the five links for which Kohn could not compute the unlinking number (recently, Nagel and Owens [12] have settled the unlinking numbers of these links).

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 Crossing change on Khovanov homology and a categorified Vassiliev skein relation

2020 ◽  
Vol 29 (07) ◽  
pp. 2050051
Author(s):  
Noboru Ito ◽  
Jun Yoshida
Keyword(s):  
Jones Polynomial ◽  
Khovanov Homology ◽  
Quantum Invariants ◽  
Quantum Invariant ◽  
Finite Type Invariants ◽  
Reidemeister Moves ◽  
Jones Polynomials ◽  
Genus One ◽  
Knots And Links ◽  
Skein Relation

Khovanov homology is a categorification of the Jones polynomial, so it may be seen as a kind of quantum invariant of knots and links. Although polynomial quantum invariants are deeply involved with Vassiliev (aka. finite type) invariants, the relation remains unclear in case of Khovanov homology. Aiming at it, in this paper, we discuss a categorified version of Vassiliev skein relation on Khovanov homology. More precisely, we will show that the “genus-one” operation gives rise to a crossing change on Khovanov complexes. Invariance under Reidemeister moves turns out, and it enables us to extend Khovanov homology to singular links. We then see that a long exact sequence of Khovanov homology groups categorifies Vassiliev skein relation for the Jones polynomials. In particular, the Jones polynomial is recovered even for singular links. We in addition discuss the FI relation on Khovanov homology.

A NOTE ON THE UTILITY OF KAUFFMAN'S STATE SUMMATION FORMULA FOR THE BRACKET POLYNOMIAL

2002 ◽  
Vol 11 (01) ◽  
pp. 13-79
Author(s):  
PAUL ISIHARA ◽  
ANDREA KOENIGSBERG
Keyword(s):  
Jones Polynomial ◽  
Summation Formula ◽  
The State ◽  
Special Method ◽  
Symbolic Algebra ◽  
Crucial Information ◽  
Bracket Polynomial ◽  
Bracket Polynomials ◽  
Knots And Links ◽  
Insight Into

Kauffman's state summation formula is especially useful for computing the bracket polynomials of projection diagrams which are related by smoothings or crossing changes. This facilitates the writing of a symbolic algebra program which computes the normalized bracket polynomials and frequencies of knots and links whose projection diagrams result from a given knot's oriented projection diagram either by crossing changes or by orientation preserving smoothings called natural smoothings. These frequencies provide insight into the unknotting game (and similar resultant games) whose object is to specify crossing changes or natural smoothings that will transform a given projection diagram of a knot into a projection diagram representing an unknot (or some other specified knot or link). The practical utility of the state summation formula is greatly enhanced by means of diagrams for closed tangle sums. These diagrams offer a special cost-reducing method to obtain crucial information needed to compute the state summation formula. This special method also gives insight into why the bracket is unchanged by mutation and contributes a strategy to the enigmatic search for a non-trivial knot with Jones polynomial equal to one.

scholarly journals Stuck Knots

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1558
Author(s):  
Khaled Bataineh
Keyword(s):  
Jones Polynomial ◽  
The Other ◽  
Natural Class ◽  
Knots And Links ◽  
Singular Knots

Singular knots and links have projections involving some usual crossings and some four-valent rigid vertices. Such vertices are symmetric in the sense that no strand overpasses the other. In this research we introduce stuck knots and links to represent physical knots and links with projections involving some stuck crossings, where the physical strands get stuck together showing which strand overpasses the other at a stuck crossing. We introduce the basic elements of the theory and we give some isotopy invariants of such knots including invariants which capture the chirality (mirror imaging) of such objects. We also introduce another natural class of stuck knots, which we call relatively stuck knots, where each stuck crossing has a stuckness factor that indicates to the value of stuckness at that crossing. Amazingly, a generalized version of Jones polynomial makes an invariant of such quantized knots and links. We give applications of stuck knots and links and their invariants in modeling and understanding bonded RNA foldings, and we explore the topology of such objects with invariants involving multiplicities at the bonds. Other perspectives are also discussed.

INVARIANTS OF SURFACE LINKS IN ℝ4 VIA SKEIN RELATION

2008 ◽  
Vol 17 (04) ◽  
pp. 439-469 ◽  
Author(s):  
SANG YOUL LEE
Keyword(s):  
Jones Polynomial ◽  
Closed Surfaces ◽  
Ambient Isotopy ◽  
Knots And Links ◽  
Skein Relation

Using Yoshikawa's surface diagram, we constructed new invariants of ambient isotopy classes of smoothly embedded closed surfaces in ℝ4 via a state-sum model similar to the Kauffman's state-sum model for the Jones polynomial for classical knots and links in ℝ3. It is shown that the invariants can also be defined by skein relation and thus they are calculated from Yoshikawa's surface diagrams recurrently. Some of the properties of the invariants are given and explicit computations for several surfaces are included.

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