scholarly journals Planar algebras and the decategorification of bordered Khovanov homology

2017 ◽  
Vol 26 (04) ◽  
pp. 1750023
Author(s):  
Lawrence P. Roberts
Keyword(s):  
Recent Work ◽  
Jones Polynomial ◽  
Khovanov Homology ◽  
Planar Surface ◽  
Simple Process ◽  
Planar Algebras ◽  
Type D ◽  
Combinatorial Construction ◽  
Planar Algebra ◽  
Compact Planar

We give a simple, combinatorial construction of a unital, spherical, non-degenerate *-planar algebra over the ring [Formula: see text]. This planar algebra is similar in spirit to the Temperley–Lieb planar algebra, but computations show that they are different. The construction comes from the combinatorics of the decategorifications of the type A and type D structures in the author’s previous work on bordered Khovanov homology. In particular, the construction illustrates how gluing of tangles occurs in the bordered Khovanov homology and its difference from that in Khovanov’s tangle homology without being encumbered by any extra homological algebra. It also provides a simple framework for showing that these theories are not related through a simple process, thereby confirming recent work of Manion. Furthermore, using Khovanov’s conventions and a state sum approach to the Jones polynomial, we obtain new invariant for tangles in [Formula: see text] where [Formula: see text] is a compact, planar surface with boundary, and the tangle intersects each boundary cylinder in an even number of points. This construction naturally generalizes Khovanov’s approach to the Jones polynomial.

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 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 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 Planar algebras, quantum information theory and subfactors

2020 ◽  
pp. 2050124
Author(s):  
Vijay Kodiyalam ◽  
Sruthymurali ◽  
V. S. Sunder
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Hadamard Matrices ◽  
Quantum Information Theory ◽  
Latin Squares ◽  
Planar Algebras ◽  
Planar Algebra

We define generalized notions of biunitary elements in planar algebras and show that objects arising in quantum information theory such as Hadamard matrices, quantum Latin squares and unitary error bases are all given by biunitary elements in the spin planar algebra. We show that there are natural subfactor planar algebras associated with biunitary elements.

scholarly journals Affine Modules and the Drinfeld Center

2016 ◽  
Vol 118 (1) ◽  
pp. 119 ◽  
Author(s):  
Paramita Das ◽  
Shamindra Kumar Ghosh ◽  
Ved Prakash Gupta
Keyword(s):  
Finite Index ◽  
Planar Algebras ◽  
Infinite Depth ◽  
Zero Level ◽  
Fusion Algebra ◽  
Planar Algebra ◽  
The Way

Given a finite index subfactor, we show that the affine morphisms at zero level in the affine category over the planar algebra associated to the subfactor is isomorphic to the fusion algebra of the subfactor as a $*$-algebra. This identification paves the way to analyze the structure of affine $P$-modules with weight zero for any subfactor planar algebra $P$ (possibly having infinite depth). Further, for irreducible depth two subfactor planar algebras, we establish an additive equivalence between the category of affine $P$-modules and the center of the category of $N$-$N$-bimodules generated by $L^2(M)$; this partially verifies a conjecture of Jones and Walker.

scholarly journals Two Lectures on the Jones Polynomial and Khovanov Homology

2017 ◽  
Author(s):  
Edward Witten
Keyword(s):  
Gauge Theory ◽  
Three Dimensional ◽  
Jones Polynomial ◽  
Khovanov Homology ◽  
Theory Approach ◽  
Laurent Polynomials ◽  
Four Dimensions ◽  
Knot Invariants ◽  
Fifth Dimension ◽  
Instanton Number

In the first of these two lectures I describe a gauge theory approach to understanding quantum knot invariants as Laurent polynomials in a complex variable q. The two main steps are to reinterpret three-dimensional Chern-Simons gauge theory in four dimensional terms and then to apply electric-magnetic duality. The variable q is associated to instanton number in the dual description in four dimensions. In the second lecture, I describe how Khovanov homology can emerge upon adding a fifth dimension.

scholarly journals Free oriented extensions of subfactor planar algebras

2018 ◽  
Vol 29 (13) ◽  
pp. 1850093 ◽  
Author(s):  
Shamindra Kumar Ghosh ◽  
Corey Jones ◽  
B. Madhav Reddy
Keyword(s):  
Left Adjoint ◽  
Planar Algebras ◽  
Planar Algebra ◽  
Restriction Functor

We show that the restriction functor from oriented factor planar algebras to subfactor planar algebras admits a left adjoint, which we call the free oriented extension functor. We show that for any subfactor planar algebra realized as the standard invariant of a hyperfinite [Formula: see text] subfactor, the projection category of the free oriented extension admits a realization as bimodules of the hyperfinite [Formula: see text] factor.

scholarly journals Khovanov homology for alternating tangles

2014 ◽  
Vol 23 (02) ◽  
pp. 1450013 ◽  
Author(s):  
Dror Bar-Natan ◽  
Hernando Burgos-Soto
Keyword(s):  
Khovanov Homology ◽  
Single Crossing ◽  
Planar Algebra ◽  
Diagonal Condition ◽  
Alternating Links ◽  
Alternating Link

We describe a "concentration on the diagonal" condition on the Khovanov complex of tangles, show that this condition is satisfied by the Khovanov complex of the single crossing tangles [Formula: see text] and [Formula: see text], and prove that it is preserved by alternating planar algebra compositions. Hence, this condition is satisfied by the Khovanov complex of all alternating tangles. Finally, in the case of 0-tangles, meaning links, our condition is equivalent to a well-known result [E. S. Lee, The support of the Khovanov's invariants for alternating links, preprint (2002), arXiv:math.GT/0201105v1.] which states that the Khovanov homology of a non-split alternating link is supported on two diagonals. Thus our condition is a generalization of Lee's theorem to the case of tangles.

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