scholarly journals Khovanov Homology of Three-Strand Braid Links

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 720
Author(s):  
Young Kwun ◽  
Abdul Nizami ◽  
Mobeen Munir ◽  
Zaffar Iqbal ◽  
Dishya Arshad ◽  
...  
Keyword(s):  
Euler Characteristic ◽  
Jones Polynomial ◽  
Khovanov Homology ◽  
Homology Groups ◽  
Link Invariants ◽  
Chain Complexes ◽  
Chain Homotopy

Khovanov homology is a categorication of the Jones polynomial. It consists of graded chain complexes which, up to chain homotopy, are link invariants, and whose graded Euler characteristic is equal to the Jones polynomial of the link. In this article we give some Khovanov homology groups of 3-strand braid links Δ 2 k + 1 = x 1 2 k + 2 x 2 x 1 2 x 2 2 x 1 2 ⋯ x 2 2 x 1 2 x 1 2 , Δ 2 k + 1 x 2 , and Δ 2 k + 1 x 1 , where Δ is the Garside element x 1 x 2 x 1 , and which are three out of all six classes of the general braid x 1 x 2 x 1 x 2 ⋯ with n factors.

A CATEGORIFICATION FOR THE PENROSE POLYNOMIAL

2011 ◽  
Vol 20 (01) ◽  
pp. 141-157
Author(s):  
KERRY LUSE ◽  
YONGWU RONG
Keyword(s):  
Euler Characteristic ◽  
Jones Polynomial ◽  
Homology Groups ◽  
Tutte Polynomials

Given a graph, we construct homology groups whose Euler characteristic is the Penrose polynomial of the graph, evaluated at an integer. This work is motivated by Khovanov's work on the categorification of the Jones polynomial for knots, and the subsequent categorifications of the chromatic and Tutte polynomials for graphs.

scholarly journals THE CATEGORIFICATION OF THE KAUFFMAN BRACKET SKEIN MODULE OF

2013 ◽  
Vol 88 (3) ◽  
pp. 407-422
Author(s):  
BOŠTJAN GABROVŠEK
Keyword(s):  
Euler Characteristic ◽  
Jones Polynomial ◽  
Chain Complex ◽  
Homology Theory ◽  
Khovanov Homology ◽  
Nonorientable Surface ◽  
Skein Module ◽  
Kauffman Bracket ◽  
Kauffman Bracket Skein Module

AbstractKhovanov homology, an invariant of links in ${ \mathbb{R} }^{3} $, is a graded homology theory that categorifies the Jones polynomial in the sense that the graded Euler characteristic of the homology is the Jones polynomial. Asaeda et al. [‘Categorification of the Kauffman bracket skein module of $I$-bundles over surfaces’, Algebr. Geom. Topol. 4 (2004), 1177–1210] generalised this construction by defining a double graded homology theory that categorifies the Kauffman bracket skein module of links in $I$-bundles over surfaces, except for the surface $ \mathbb{R} {\mathrm{P} }^{2} $, where the construction fails due to strange behaviour of links when projected to the nonorientable surface $ \mathbb{R} {\mathrm{P} }^{2} $. This paper categorifies the missing case of the twisted $I$-bundle over $ \mathbb{R} {\mathrm{P} }^{2} $, $ \mathbb{R} {\mathrm{P} }^{2} \widetilde {\times } I\approx \mathbb{R} {\mathrm{P} }^{3} \setminus \{ \ast \} $, by redefining the differential in the Khovanov chain complex in a suitable manner.

scholarly journals ALL LINK INVARIANTS FOR TWO DIMENSIONAL SOLUTIONS OF YANG–BAXTER EQUATION AND DRESSINGS

2006 ◽  
Vol 15 (10) ◽  
pp. 1279-1301
Author(s):  
N. AIZAWA ◽  
M. HARADA ◽  
M. KAWAGUCHI ◽  
E. OTSUKI
Keyword(s):  
Jones Polynomial ◽  
Alexander Polynomial ◽  
Baxter Equation ◽  
Two Dimensional ◽  
Link Invariant ◽  
Three Solutions ◽  
Polynomial Invariants ◽  
Link Invariants ◽  
Higher Dimensional

All polynomial invariants of links for two dimensional solutions of Yang–Baxter equation is constructed by employing Turaev's method. As a consequence, it is proved that the best invariant so constructed is the Jones polynomial and there exist three solutions connecting to the Alexander polynomial. Invariants for higher dimensional solutions, obtained by the so-called dressings, are also investigated. It is observed that the dressings do not improve link invariant unless some restrictions are put on dressed solutions.

scholarly journals Euler characteristics and their congruences in the positive rank setting

2020 ◽  
pp. 1-18
Author(s):  
Anwesh Ray ◽  
R. Sujatha
Keyword(s):  
Elliptic Curves ◽  
Euler Characteristic ◽  
Selmer Groups ◽  
Euler Characteristics ◽  
Rank Zero ◽  
Homology Groups ◽  
Main Conjecture ◽  
Congruence Relations ◽  
Higher Rank ◽  
Positive Rank

Abstract The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible p-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves. The results provide evidence for the p-adic Birch and Swinnerton-Dyer formula without assuming the main conjecture.

A generalized skein relation for Khovanov homology and a categorification of the θ-invariant

2020 ◽  
pp. 1-27
Author(s):  
M. Chlouveraki ◽  
D. Goundaroulis ◽  
A. Kontogeorgis ◽  
S. Lambropoulou
Keyword(s):  
Jones Polynomial ◽  
Khovanov Homology ◽  
Spectral Sequences ◽  
Link Invariant ◽  
Type Relation ◽  
Skein Relation

The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type relation satisfied by the Khovanov homology. Thanks to this relation, we are able to generalize the Khovanov homology in order to obtain a categorification of the θ-invariant, which is itself a generalization of the Jones polynomial.

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 The decategorification of bordered Khovanov homology

2014 ◽  
Vol 23 (14) ◽  
pp. 1450078 ◽  
Author(s):  
Lawrence P. Roberts
Keyword(s):  
Floer Homology ◽  
Jones Polynomial ◽  
Type A ◽  
Khovanov Homology ◽  
Heegaard Floer Homology ◽  
Type D ◽  
Heegaard Floer

In [A type D structure in Khovanov homology, preprint (2013), arXiv:1304.0463; A type A structure in Khovanov homology, preprint (2013), arXiv:1304.0465] the author constructed a package which describes how to decompose the Khovanov homology of a link ℒ into the algebraic pairing of a type D structure and a type A structure (as defined in bordered Floer homology), whenever a diagram for ℒ is decomposed into the union of two tangles. Since Khovanov homology is the categorification of a version of the Jones polynomial, it is natural to ask what the types A and D structures categorify, and how their pairing is encoded in the decategorifications. In this paper, the author constructs the decategorifications of these two structures, in a manner similar to I. Petkova's decategorification of bordered Floer homology, [The decategorification of bordered Heegaard-Floer homology, preprint (2012), arXiv:1212.4529v1], and shows how they recover the Jones polynomial. We also use the decategorifications to compare this approach to tangle decompositions with M. Khovanov's from [A functor-valued invariant of tangles, Algebr. Geom. Topol.2 (2002) 665–741].

scholarly journals CATEGORIFICATION OF THE DICHROMATIC POLYNOMIAL FOR GRAPHS

2008 ◽  
Vol 17 (01) ◽  
pp. 31-45 ◽  
Author(s):  
MARKO STOŠIĆ
Keyword(s):  
Positive Integer ◽  
Euler Characteristic ◽  
Jones Polynomial ◽  
Chain Complex ◽  
Tutte Polynomial ◽  
A Chain ◽  
Alternating Link ◽  
Dichromatic Polynomial

For each graph and each positive integer n, we define a chain complex whose graded Euler characteristic is equal to an appropriate n-specialization of the dichromatic polynomial. This also gives a categorification of n-specializations of the Tutte polynomial of graphs. Also, for each graph and integer n ≤ 2, we define the different one-variable n-specializations of the dichromatic polynomial, and for each polynomial, we define graded chain complex whose graded Euler characteristic is equal to that polynomial. Furthermore, we explicitly categorify the specialization of the Tutte polynomial for graphs which corresponds to the Jones polynomial of the appropriate alternating link.

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