scholarly journals The Jones polynomial of pretzel knots and links

1998 ◽  
Vol 83 (2) ◽  
pp. 135-147 ◽  
Author(s):  
Ryan A. Landvoy
Keyword(s):  
Jones Polynomial ◽  
Pretzel Knots ◽  
Knots And Links

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 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 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.

Sliceness of alternating pretzel knots and links

2020 ◽  
Vol 282 ◽  
pp. 107317
Author(s):  
Kengo Kishimoto ◽  
Tetsuo Shibuya ◽  
Tatsuya Tsukamoto
Keyword(s):  
Pretzel Knots ◽  
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].

ON 2-ADJACENCY OF CLASSICAL PRETZEL KNOTS

2013 ◽  
Vol 22 (11) ◽  
pp. 1350066 ◽  
Author(s):  
ZHI-XIONG TAO
Keyword(s):  
Jones Polynomial ◽  
Conway Polynomial ◽  
Pretzel Knots ◽  
Trefoil Knot ◽  
Figure Eight Knot

For classical (3-strand) pretzel knots (including even type and odd type), we study their 2-adjacency using Conway polynomial and Jones polynomial. We show that only the trefoil knot and the figure-eight knot in these knots are 2-adjacent.

scholarly journals Colored HOMFLY polynomials for the pretzel knots and links

2015 ◽  
Vol 2015 (7) ◽  
Author(s):  
A. Mironov ◽  
A. Morozov ◽  
A. Sleptsov
Keyword(s):  
Pretzel Knots ◽  
Homfly Polynomials ◽  
Knots And Links

Identifying non-pseudo-alternating knots by using the free factor property of minimal genus Seifert surfaces

2021 ◽  
Author(s):  
Keisuke Himeno ◽  
Masakazu Teragaito
Keyword(s):  
Seifert Surface ◽  
Major Difficulty ◽  
Flat Surfaces ◽  
Minimal Genus ◽  
Pretzel Knots ◽  
Seifert Surfaces ◽  
Manifold Theory ◽  
Knots And Links

Pseudo-alternating knots and links are defined constructively via their Seifert surfaces. By performing Murasugi sums of primitive flat surfaces, such a knot or link is obtained as the boundary of the resulting surface. Conversely, it is hard to determine whether a given knot or link is pseudo-alternating or not. A major difficulty is the lack of criteria to recognize whether a given Seifert surface is decomposable as a Murasugi sum. In this paper, we propose a new idea to identify non-pseudo-alternating knots. Combining with the uniqueness of minimal genus Seifert surface obtained through sutured manifold theory, we demonstrate that two infinite classes of pretzel knots are not pseudo-alternating.

scholarly journals Colored knot polynomials for arbitrary pretzel knots and links

2015 ◽  
Vol 743 ◽  
pp. 71-74 ◽  
Author(s):  
D. Galakhov ◽  
D. Melnikov ◽  
A. Mironov ◽  
A. Morozov ◽  
A. Sleptsov
Keyword(s):  
Pretzel Knots ◽  
Knots And 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.

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