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 KHOVANOV HOMOLOGY AND THE TWIST NUMBER OF ALTERNATING KNOTS

2009 ◽  
Vol 18 (12) ◽  
pp. 1651-1662
Author(s):  
ROBERT G. TODD
Keyword(s):  
Exact Sequence ◽  
Jones Polynomial ◽  
Khovanov Homology ◽  
Absolute Value ◽  
Kauffman Bracket ◽  
The Absolute ◽  
Twist Number ◽  
Skein Relation

In A Volumish Theorem for the Jones Polynomial, by O. Dasbach and X. S. Lin, it was shown that the sum of the absolute value of the second and penultimate coefficient of the Jones polynomial of an alternating knot is equal to the twist number of the knot. Here we give a new proof of this result using Khovanov homology. The proof is by induction on the number of crossings using the long exact sequence in Khovanov homology corresponding to the Kauffman bracket skein relation.

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.

scholarly journals AN INVARIANT FOR SINGULAR KNOTS

2009 ◽  
Vol 18 (06) ◽  
pp. 825-840 ◽  
Author(s):  
J. JUYUMAYA ◽  
S. LAMBROPOULOU
Keyword(s):  
Hecke Algebras ◽  
Quadratic Relation ◽  
Link Invariant ◽  
Topological Interpretation ◽  
Type Invariant ◽  
Singular Link ◽  
Markov Trace ◽  
Skein Relation ◽  
Singular Knots

In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma–Hecke algebras Y d,n(u) and the theory of singular braids. The Yokonuma–Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid SBn into the algebra Y d,n(u). Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relation involving singular crossings, which arises from a quadratic relation in the algebra Y d,n(u).

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 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.

scholarly journals On spectral sequences from Khovanov homology

2020 ◽  
Vol 20 (2) ◽  
pp. 531-564
Author(s):  
Andrew Lobb ◽  
Raphael Zentner
Keyword(s):  
Khovanov Homology ◽  
Spectral Sequences

scholarly journals The colored Kauffman Skein relation and the head and tail of the colored Jones polynomial

2017 ◽  
Vol 26 (03) ◽  
pp. 1741002 ◽  
Author(s):  
Mustafa Hajij
Keyword(s):  
Natural Extension ◽  
Jones Polynomial ◽  
Colored Jones Polynomial ◽  
Alternating Links ◽  
Skein Relation

Using the colored Kauffman skein relation, we study the highest and lowest [Formula: see text] coefficients of the [Formula: see text] unreduced colored Jones polynomial of alternating links. This gives a natural extension of a result by Kauffman in regard with the Jones polynomial of alternating links and its highest and lowest coefficients. We also use our techniques to give a new and natural proof for the existence of the tail of the colored Jones polynomial for alternating links.

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].

Quantum algorithms for the Jones polynomial and Khovanov homology

2012 ◽  
Author(s):  
Louis H. Kauffman ◽  
Samuel J. Lomonaco
Keyword(s):  
Quantum Algorithms ◽  
Jones Polynomial ◽  
Khovanov Homology
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