scholarly journals The rational Khovanov homology of 3-strand pretzel links

2014 ◽  
Vol 23 (08) ◽  
pp. 1450040 ◽  
Author(s):  
Andrew Manion
Keyword(s):  
General Formula ◽  
Knot Theory ◽  
Rational Numbers ◽  
Khovanov Homology ◽  
Pretzel Knots ◽  
Knots And Links

The 3-strand pretzel knots and links are a well-studied source of examples in knot theory. However, while there have been computations of the Khovanov homology and Rasmussen s-invariants of some sub-families of 3-strand pretzel knots, no general formula has been given for all of them. We give a formula for the unreduced Khovanov homology, over the rational numbers, of all 3-strand pretzel links. We also compute generalized s-invariants of these links.

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 BAND SURGERY ON KNOTS AND LINKS

2010 ◽  
Vol 19 (12) ◽  
pp. 1535-1547 ◽  
Author(s):  
TAIZO KANENOBU
Keyword(s):  
Knot Theory ◽  
Site Specific ◽  
Site Specific Recombination ◽  
Knots And Links

We give some relationships of the Jones and Q polynomials between two links which are related by a band surgery. Then we consider two applications: The first one is to an evaluation of the ribbon-fusion number, the least fusion number of a ribbon knot. The second one is to DNA knot theory, helping us to understand the action of the Xer site-specific recombination at psi site.

scholarly journals A VOLUME FORM ON THE KHOVANOV INVARIANT

2012 ◽  
Vol 21 (04) ◽  
pp. 1250032
Author(s):  
JUAN ORTIZ-NAVARRO
Keyword(s):  
Chain Complex ◽  
Khovanov Homology ◽  
Volume Form ◽  
Reidemeister Torsion ◽  
Homology Groups ◽  
Numerical Invariant ◽  
Reidemeister Moves ◽  
Invariant Volume ◽  
Knots And Links

The Reidemeister torsion construction can be applied to the chain complex used to compute the Khovanov homology of a knot or a link. This defines a volume form on Khovanov homology. The volume form transforms correctly under Reidemeister moves to give an invariant volume on the Khovanov homology. In this paper, its construction and invariance under these moves is demonstrated. Also, some examples of the invariant are presented for particular choices for the bases of homology groups to obtain a numerical invariant of knots and links. In these examples, the algebraic torsion seen in the Khovanov chain complex when homology is computed over ℤ is recovered.

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.

scholarly journals PATTERNS IN ODD KHOVANOV HOMOLOGY

2011 ◽  
Vol 20 (01) ◽  
pp. 203-222 ◽  
Author(s):  
ALEXANDER N. SHUMAKOVITCH
Keyword(s):  
Knot Theory ◽  
Khovanov Homology ◽  
Low Dimensional Topology ◽  
Low Dimensional ◽  
Compare And Contrast

We investigate properties of the odd Khovanov homology, compare and contrast them with those of the original (even) Khovanov homology, and discuss applications of the odd Khovanov homology to other areas of knot theory and low-dimensional topology.

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 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.

scholarly journals On Legendre numbers

1985 ◽  
Vol 8 (2) ◽  
pp. 407-411 ◽  
Author(s):  
Paul W. Haggard
Keyword(s):  
General Formula ◽  
Legendre Polynomials ◽  
Rational Numbers ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Infinite Set ◽  
Derivatives Of

The Legendre numbers, an infinite set of rational numbers are defined from the associated Legendre functions and several elementary properties are presented. A general formula for the Legendre numbers is given. Applications include summing certain series of Legendre numbers and evaluating certain integrals. Legendre numbers are used to obtain the derivatives of all orders of the Legendre polynomials atx=1.

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].

Sign in / Sign up

Export Citation Format

Share Document