scholarly journals OPEN-CLOSED TQFTS EXTEND KHOVANOV HOMOLOGY FROM LINKS TO TANGLES

2009 ◽  
Vol 18 (01) ◽  
pp. 87-150 ◽  
Author(s):  
AARON D. LAUDA ◽  
HENDRYK PFEIFFER
Keyword(s):  
Chain Complex ◽  
Special Kind ◽  
Homology Theory ◽  
Khovanov Homology ◽  
Quantum Field Theories ◽  
Finite Characteristic ◽  
Link Homology ◽  
Topological Quantum ◽  
Carry Over ◽  
A Chain

We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to represent a refinement of Bar-Natan's universal geometric complex algebraically, and thereby extend Khovanov homology from links to arbitrary tangles. For every plane diagram of an oriented tangle, we construct a chain complex whose terms are modules of a suitable algebra A such that there is one action of A or A op for every boundary point of the tangle. We give examples of such algebras A for which our tangle homology theory reduces to the link homology theories of Khovanov, Lee and Bar-Natan if it is evaluated for links. As a consequence of the Cardy condition, Khovanov's graded theory can only be extended to tangles if the underlying field has finite characteristic. Whenever the open-closed TQFT arises from a state-sum construction, we obtain honest planar algebra morphisms, and all composition properties of the universal geometric complex carry over to the algebraic complex. We give examples of state-sum open-closed TQFTs for which one can still determine both characteristic p Khovanov homology of links and Rasmussen's s-invariant.

scholarly journals A KHOVANOV TYPE INVARIANT DERIVED FROM AN UNORIENTED HQFT FOR LINKS IN THICKENED SURFACES

2013 ◽  
Vol 24 (10) ◽  
pp. 1350078 ◽  
Author(s):  
KEIJI TAGAMI
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Equivalence Class ◽  
Chain Complex ◽  
Quantum Field ◽  
Link Homology ◽  
Stable Equivalence ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Topological Quantum

Two link diagrams on compact surfaces are strongly equivalent if they are related by Reidemeister moves and orientation preserving homeomorphisms of the surfaces. They are stably equivalent if they are related by the two previous operations and adding or removing handles. Turaev and Turner constructed a link homology for each stable equivalence class by applying an unoriented topological quantum field theory (TQFT) to a geometric chain complex similar to Bar-Natan's one. In this paper, by using an unoriented homotopy quantum field theory (HQFT), we construct a link homology for each strong equivalence class. Moreover, our homology yields an invariant of links in the oriented I-bundle of a compact surface.

scholarly journals CALCULATING BAR-NATAN'S CHARACTERISTIC TWO KHOVANOV HOMOLOGY

2006 ◽  
Vol 15 (10) ◽  
pp. 1335-1356 ◽  
Author(s):  
PAUL R. TURNER
Keyword(s):  
Homology Theory ◽  
Khovanov Homology ◽  
Spectral Sequences ◽  
Link Homology ◽  
Characteristic Two

We investigate Bar-Natan's characteristic two Khovanov link homology theory studying both the filtered and bi-graded theories. The filtered theory is computed explicitly and the bi-graded theory analysed by setting up a family of spectral sequences. The E2-pages can be described in terms of groups arising from the action of a certain endomorphism on 𝔽2-Khovanov homology. Some simple consequences are discussed.

ON CONJECTURES ABOUT POSITIVE BRAID KNOTS AND ALMOST ALTERNATING TORUS KNOTS

2010 ◽  
Vol 19 (11) ◽  
pp. 1471-1486
Author(s):  
MARKO STOŠIĆ
Keyword(s):  
Positive Integer ◽  
Homology Group ◽  
Knot Theory ◽  
Homology Theory ◽  
Khovanov Homology ◽  
Torus Knots ◽  
Link Homology

In this paper we resolve some conjectures concerning positive braid knots and almost alternating torus knots. Namely, we prove that the first Khovanov homology group of positive braid knot is trivial, as conjectured by Khovanov. Also, we generalize this result to show that the same is true in the case of Khovanov–Rozansky homology (sl(n) link homology) for any positive integer n. Moreover, by using the Khovanov homology theory, we prove the classical knot theory conjecture by Adams, that the only almost alternating torus knots are T3, 4 and T3, 5.

Twisted skein homology

2014 ◽  
Vol 23 (05) ◽  
pp. 1450027
Author(s):  
Nguyen D. Duong ◽  
Lawrence P. Roberts
Keyword(s):  
Chain Complex ◽  
Khovanov Homology ◽  
A Chain ◽  
Orientable Surfaces

We apply the techniques of totally twisted Khovanov homology to Asaeda, Przytycki, and Sikora's construction of Khovanov type homologies for links and tangles in I-bundles over (orientable) surfaces. As a result we describe a chain complex built out of resolutions with only noncontractible circles whose homology is an invariant of the tangle. We use these to understand the δ-graded homology for links with alternating diagrams in the surface.

scholarly journals Stable homotopy refinement of quantum annular homology

2021 ◽  
Vol 157 (4) ◽  
pp. 710-769
Author(s):  
Rostislav Akhmechet ◽  
Vyacheslav Krushkal ◽  
Michael Willis
Keyword(s):  
Structural Properties ◽  
Cyclic Group ◽  
Group Action ◽  
Homology Theory ◽  
Khovanov Homology ◽  
Stable Homotopy ◽  
Spectrum Level ◽  
Link Homology ◽  
Cyclic Group Action ◽  
Equivariant Version

We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq ~2$ we associate to an annular link $L$ a naive $\mathbb {Z}/r\mathbb {Z}$ -equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as modules over $\mathbb {Z}[\mathbb {Z}/r\mathbb {Z}]$ . The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology.

Bar-Natan’s geometric complex and a categorification of the Dye–Kauffman–Miyazawa polynomial

2016 ◽  
Vol 25 (01) ◽  
pp. 1550076
Author(s):  
Keiji Tagami
Keyword(s):  
Equivalence Class ◽  
Euler Characteristic ◽  
Khovanov Homology ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Link Homology ◽  
Stable Equivalence ◽  
Special Case

In this paper, we construct a categorification of the two-variable Dye–Kauffman–Miyazawa polynomial by utilizing Bar-Natan’s construction of the Khovanov homology and homotopy quantum field theories (HQFTs) given by Turaev. In particular, for any stable equivalence class, we construct a [Formula: see text]-graded link homology over [Formula: see text] whose graded Euler characteristic is the two-variable Dye–Kauffman–Miyazawa polynomial. Moreover, we show that it is isomorphic to a special case of Dye–Kauffman–Manturov’s categorification. In this sense, we explain the special case of Dye–Kauffman–Manturov’s homology in terms of Bar-Natan’s construction.

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 Domain walls in 4d $$ \mathcal{N} $$ = 1 SYM

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Diego Delmastro ◽  
Jaume Gomis
Keyword(s):  
Domain Wall ◽  
Domain Walls ◽  
Simons Theory ◽  
Partition Functions ◽  
Quantum Field Theories ◽  
Yang Mills ◽  
Stable Domain ◽  
Coxeter Number ◽  
Topological Quantum ◽  
Simply Connected

Abstract 4d$$ \mathcal{N} $$ N = 1 super Yang-Mills (SYM) with simply connected gauge group G has h gapped vacua arising from the spontaneously broken discrete R-symmetry, where h is the dual Coxeter number of G. Therefore, the theory admits stable domain walls interpolating between any two vacua, but it is a nonperturbative problem to determine the low energy theory on the domain wall. We put forward an explicit answer to this question for all the domain walls for G = SU(N), Sp(N), Spin(N) and G2, and for the minimal domain wall connecting neighboring vacua for arbitrary G. We propose that the domain wall theories support specific nontrivial topological quantum field theories (TQFTs), which include the Chern-Simons theory proposed long ago by Acharya-Vafa for SU(N). We provide nontrivial evidence for our proposals by exactly matching renormalization group invariant partition functions twisted by global symmetries of SYM computed in the ultraviolet with those computed in our proposed infrared TQFTs. A crucial element in this matching is constructing the Hilbert space of spin TQFTs, that is, theories that depend on the spin structure of spacetime and admit fermionic states — a subject we delve into in some detail.

GENERAL COVARIANCE, TOPOLOGICAL QUANTUM FIELD THEORIES AND FRACTIONAL STATISTICS

1992 ◽  
Vol 07 (02) ◽  
pp. 209-234 ◽  
Author(s):  
J. GAMBOA
Keyword(s):  
Hamiltonian Formalism ◽  
Field Theories ◽  
General Covariance ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Fractional Statistics ◽  
Multiply Connected ◽  
Topological Quantum ◽  
Gauge Fixing ◽  
Topological Quantum Field Theories

Topological quantum field theories and fractional statistics are both defined in multiply connected manifolds. We study the relationship between both theories in 2 + 1 dimensions and we show that, due to the multiply-connected character of the manifold, the propagator for any quantum (field) theory always contains a first order pole that can be identified with a physical excitation with fractional spin. The article starts by reviewing the definition of general covariance in the Hamiltonian formalism, the gauge-fixing problem and the quantization following the lines of Batalin, Fradkin and Vilkovisky. The BRST–BFV quantization is reviewed in order to understand the topological approach proposed here.

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