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.

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.

Strongly spin-preserving solutions of the Yang-Baxter equation and their link invariants

1993 ◽  
Vol 113 (2) ◽  
pp. 401-411
Author(s):  
M. A. Hennings
Keyword(s):  
Jones Polynomial ◽  
Alexander Polynomial ◽  
The Other ◽  
Baxter Equation ◽  
Link Invariants

Recently Kauffman[2, 3] has classified the (strongly) spin-preserving solutions of the Yang—Baxter equation, and in particular discussed two solutions. One of these can be used to obtain the Jones polynomial, while the other (with care) leads to the Alexander polynomial. In this paper we shall complete this analysis, and shall describe all strongly spin-preserving invertible solutions of the Yang-Baxter equation which lead to link invariants. Having done this, we investigate what sort of link invariants can be obtained from these solutions. It turns out that these invariants are precisely those which have been described in Hennings [l].

scholarly journals The Influence of Guttural Consonants /χ/, /ħ/, and /h/ on Vowel /a/ in Saudi Arabic

2019 ◽  
Vol 10 (1) ◽  
pp. 115
Author(s):  
Haroon N. Alsager
Keyword(s):  
Comparative Study ◽  
Acoustic Analysis ◽  
Unified Model ◽  
The Other ◽  
Analysis Method ◽  
Formant Frequency ◽  
Natural Class ◽  
Arabic Speakers ◽  
Second Formant ◽  
Saudi Arabic

This paper presents a comparative study which investigates the influence of Saudi Arabic guttural consonants /χ/, /ħ/ and /h/ on the vowel /a/ when they are adjacent and in the same syllable. Cohn (2007, 2009), Flemming (2001), and Keating (1996) discuss a unified model in which phonology and phonetics are treated as two distinct elements of one domain where each element has an effect on the other to some degree. McCarthy (1991, 1994), Rose (1996), Zawaydeh (1999, 2004), and BinMuqbil (2006) presented phonological studies on gutturals, as well as discussions on gutturals as a natural class, which uphold the phonological aspect of Cohn’s (2009) unified model. The aim of this study is to address the phonetic aspect of Cohn’s (2009) unified model by analyzing the phonetic effects of guttural-vowel coarticulation. An acoustic analysis method was used as a framework for this investigation to extract first formant frequency (F1) and second formant frequency (F2) to measure the influence in the coarticulation. For the purpose of this study, seven native Saudi Arabic speakers were recorded pronouncing 70 Saudi Arabic words. The results showed that guttural consonants have an influence on the vowel /a/ by lowering and backing it when they are adjacent and in the same syllable, while the vowel /a/ in the nonguttural consonants is raising and fronting their adjacent vowel /a/ in the same syllable in comparison with the vowel /a/ in the guttural environment.

scholarly journals Singular knots and involutive quandles

2017 ◽  
Vol 26 (14) ◽  
pp. 1750099 ◽  
Author(s):  
Indu R. U. Churchill ◽  
Mohamed Elhamdadi ◽  
Mustafa Hajij ◽  
Sam Nelson
Keyword(s):  
Knot Theory ◽  
Algebraic Structures ◽  
Reidemeister Moves ◽  
Knots And Links ◽  
Singular Knots

The aim of this paper is to define certain algebraic structures coming from generalized Reidemeister moves of singular knot theory. We give examples, show that the set of colorings by these algebraic structures is an invariant of singular links. As an application, we distinguish several singular knots and links.

The Tait conjecture in S1 × S2

2016 ◽  
Vol 25 (11) ◽  
pp. 1650063 ◽  
Author(s):  
Alessio Carrega
Keyword(s):  
Analogous Result ◽  
Jones Polynomial ◽  
The Other ◽  
Complete Answer ◽  
Other Hand ◽  
Alternating Links

The Tait conjecture states that reduced alternating diagrams of links in [Formula: see text] have the minimal number of crossings. It has been proved in 1987 by M. Thistlethwaite, L. H. Kauffman and K. Murasugi studying the Jones polynomial. In this paper, we prove an analogous result for alternating links in [Formula: see text] giving a complete answer to this problem. In [Formula: see text] we find a dichotomy: the appropriate version of the statement is true for [Formula: see text]-homologically trivial links, and our proof also uses the Jones polynomial. On the other hand, the statement is false for [Formula: see text]-homologically non-trivial links, for which the Jones polynomial vanishes.

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 COLORED JONES POLYNOMIALS WITH POLYNOMIAL GROWTH

2008 ◽  
Vol 10 (supp01) ◽  
pp. 815-834 ◽  
Author(s):  
KAZUHIRO HIKAMI ◽  
HITOSHI MURAKAMI
Keyword(s):  
Irreducible Representation ◽  
Complex Number ◽  
Polynomial Growth ◽  
Jones Polynomial ◽  
The Other ◽  
Absolute Value ◽  
Volume Conjecture ◽  
Jones Polynomials ◽  
Dimensional Irreducible Representation ◽  
Colored Jones Polynomials

The volume conjecture and its generalizations say that the colored Jones polynomial corresponding to the N-dimensional irreducible representation of sl(2;ℂ) of a (hyperbolic) knot evaluated at exp(c/N) grows exponentially with respect to N if one fixes a complex number c near [Formula: see text]. On the other hand if the absolute value of c is small enough, it converges to the inverse of the Alexander polynomial evaluated at exp c. In this paper we study cases where it grows polynomially.

Sign in / Sign up

Export Citation Format

Share Document