- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 03:19:45
14
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=QT&vol=2&update_since=2024-03-29
Quantum Topology
Quantum Topol.
QT
1663-487X
1664-073X
General
10.4171/QT
http://www.ems-ph.org/doi/10.4171/QT
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
2
2011
1
Differential forms and 0-dimensional supersymmetric field theories
Henning
Hohnhold
Statistisches Bundesamt, WIESBADEN, GERMANY
Matthias
Kreck
Universität Bonn, BONN, GERMANY
Stephan
Stolz
University of Notre Dame, NOTRE DAME, UNITED STATES
Peter
Teichner
University of California, BERKELEY, UNITED STATES
Supermanifolds, differential forms, stacks, field theory, cohomology theory
We show that closed differential forms on a smooth manifold X can be interpreted as topological (respectively Eudlidean) supersymmetric field theories of dimension 0|1 over X. As a consequence, concordance classes of such field theories are shown to represent de Rham cohomology. The main contribution of this paper is to make all new mathematical notions regarding supersymmetric field theories precise.
$K$-theory
Category theory; homological algebra
Operator theory
Manifolds and cell complexes
1
41
10.4171/QT/12
http://www.ems-ph.org/doi/10.4171/QT/12
The Jones slopes of a knot
Stavros
Garoufalidis
Georgia Institute of Technology, ATLANTA, UNITED STATES
Knot, link, Jones polynomial, Jones slope, Jones period, quasi-polynomial, alternating knots, signature, pretzel knots, polytopes, Newton polygon, incompressible surfaces, slope, slope conjecture
The paper introduces the slope conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural number) of a knot in 3-space. We formulate a number of conjectures for these invariants and verify them by explicit computations for the class of alternating knots, the knots with at most 9 crossings, the torus knots and the (−2,3,n) pretzel knots.
Manifolds and cell complexes
General
43
69
10.4171/QT/13
http://www.ems-ph.org/doi/10.4171/QT/13
An intrinsic approach to invariants of framed links in 3-manifolds
Efstratia
Kalfagianni
Michigan State University, EAST LANSING, UNITED STATES
Characteristic submanifold, framed links, finite type invariants, Kauffman skein module, loop space, Seifert fibered 3-manifolds, toroidal decompositions
We study framed links in irreducible 3-manifolds that are ℤ-homology 3-spheres or atoroidal ℚ-homology 3-spheres. We calculate the dual of the Kauffman skein module over the ring of two variable power series with complex coefficients. For links in S3 we give a new construction of the classical Kauffman polynomial.
Manifolds and cell complexes
General
71
96
10.4171/QT/14
http://www.ems-ph.org/doi/10.4171/QT/14
Erratum to: “A categorification of quantum sl(n)”
Mikhail
Khovanov
Columbia University, NEW YORK, UNITED STATES
Aaron
Lauda
University of Southern California, LOS ANGELES, UNITED STATES
A sign error in Lemma 6.4 in Quantum Topol. 1 (2010), 1–92, is corrected.
Quantum theory
General
97
99
10.4171/QT/15
http://www.ems-ph.org/doi/10.4171/QT/15
2
Knot polynomial identities and quantum group coincidences
Scott
Morrison
University of California, BERKELEY, UNITED STATES
Emily
Peters
Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES
Noah
Snyder
Indiana University, BLOOMINGTON, UNITED STATES
Planar algebras, quantum groups, fusion categories, knot theory, link invariants
We construct link invariants using the $\mathcal{D}_{2n}$ subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to specializations of other quantum knot polynomials. These identities can also be explained by coincidences between small modular categories involving the even parts of the $\mathcal{D}_{2n}$ planar algebras. We discuss the origins of these coincidences, explaining the role of SO level-rank duality, Kirby–Melvin symmetry, and properties of small Dynkin diagrams. One of these coincidences involves $G_2$ and does not appear to be related to level-rank duality.
Category theory; homological algebra
Nonassociative rings and algebras
Manifolds and cell complexes
Quantum theory
101
156
10.4171/QT/16
http://www.ems-ph.org/doi/10.4171/QT/16
Universal skein theory for finite depth subfactor planar algebras
Vijay
Kodiyalam
, CHENNAI, INDIA
Srikanth
Tupurani
, CHENNAI, INDIA
Skein theory, subfactors, planar algebras
We describe an explicit finite presentation for a finite depth subfactor planar algebra. We also show that such planar algebras are singly generated with the generator subject to finitely many relations.
Functional analysis
Manifolds and cell complexes
General
157
172
10.4171/QT/17
http://www.ems-ph.org/doi/10.4171/QT/17
A quaternionic braid representation (after Goldschmidt and Jones)
Eric
Rowell
Texas A&M University, COLLEGE STATION, UNITED STATES
Braid group, Hecke algebra, modular category
We show that the braid group representations associated with the (3,6)-quotients of the Hecke algebras factor over a finite group. This was known to experts going back to the 1980s, but a proof has never appeared in print. Our proof uses an unpublished quaternionic representation of the braid group due to Goldschmidt and Jones. Possible topological and categorical generalizations are discussed.
Group theory and generalizations
Manifolds and cell complexes
General
173
182
10.4171/QT/18
http://www.ems-ph.org/doi/10.4171/QT/18
HOMFLY-PT polynomial and normal rulings of Legendrian solid torus links
Dan
Rutherford
Duke University, DURHAM, UNITED STATES
Legendrian link, HOMFLY-PT polynomial, normal rulings, symmetric functions
We show that for any Legendrian link $L$ in the 1-jet space of $S^1$ the 2-graded ruling polynomial, $R^2_L(z)$, is determined by the Thurston--Bennequin number and the HOMFLY-PT polynomial. Specifically, we recover $R^2_L(z)$ as a coefficient of a particular specialization of the HOMFLY-PT polynomial. Furthermore, we show that this specialization may be interpreted as the standard inner product on the algebra of symmetric functions that is often identified with a certain subalgebra of the HOMFLY-PT skein module of the solid torus. In contrast to the 2-graded case, we are able to use 0-graded ruling polynomials to distinguish many homotopically non-trivial Legendrian links with identical classical invariants.
Manifolds and cell complexes
General
183
215
10.4171/QT/19
http://www.ems-ph.org/doi/10.4171/QT/19
3
A note on sign conventions in link Floer homology
Sucharit
Sarkar
Princeton University, PRINCETON, UNITED STATES
Sign convention, link Floer homology, grid diagram
For knots in $S^3$, the bi-graded hat version of knot Floer homology is defined over $\mathbb{Z}$; however, for an $l$-component link $L$ in $S^3$ with $l>1$, there are $2^{l-1}$ bi-graded hat versions of link Floer homology defined over $\mathbb{Z}$; the multi-graded hat version of link Floer homology, defined from holomorphic considerations, is only defined over $\mathbb{F}_2$; and there is a multi-graded version of link Floer homology defined over $\mathbb{Z}$ using grid diagrams. In this short note, we try to address this issue, by extending the $\mathbb{F}_2$-valued multi-graded link Floer homology theory to $2^{l-1}$ $\mathbb{Z}$-valued theories. A grid diagram representing a link gives rise to a chain complex over $\mathbb{F}_2$, whose homology is related to the multi-graded hat version of link Floer homology of that link over $\mathbb{F}_2$. A sign refinement of the chain complex exists, and for knots, we establish that the sign refinement does indeed correspond to the sign assignment for the hat version of the knot Floer homology. For links, we create $2^{l-1}$ sign assignments on the grid diagrams, and show that they are related to the $2^{l-1}$ multi-graded hat versions of link Floer homology over $\mathbb{Z}$, and one of them corresponds to the existing sign refinement of the grid chain complex.
Manifolds and cell complexes
General
217
239
10.4171/QT/20
http://www.ems-ph.org/doi/10.4171/QT/20
Categorification of level two representations of quantum sln via generalized arc rings
Yanfeng
Chen
, JERSEY CITY, UNITED STATES
Categorification, invariant, tangle cobordism, quantum group, arc ring, bimodule, representation, exact functor
In this paper we construct an extension of the arc ring $H^n$ introduced by Khovanov [4], and use it to categorify level two representations of $U_q(sl_N)$. These rings also induce invariants of tangle
Manifolds and cell complexes
Associative rings and algebras
Quantum theory
General
241
267
10.4171/QT/21
http://www.ems-ph.org/doi/10.4171/QT/21
A quadrilateral in the Asaeda–Haagerup category
Marta
Asaeda
University of California, RIVERSIDE, UNITED STATES
Subfactors, fusion categories
We construct a noncommuting quadrilateral of factors whose upper sides are each the Asaeda–Haagerup subfactor with index $\frac{5+\sqrt{17}}{2} $ by showing the existence of a $Q$-system in the Asaeda–Haagerup category with index $\frac{7+\sqrt{17}}{2} $. We also conjecture the existence of a Q-system in the same category with index $\frac{9+ \sqrt{17}}{2} $ and an associated quadrilateral whose upper sides have index $\frac{7+\sqrt{17}}{2} $.
Functional analysis
General
269
300
10.4171/QT/22
http://www.ems-ph.org/doi/10.4171/QT/22
The embedding theorem for finite depth subfactor planar algebras
Vaughan
Jones
Vanderbilt University, NASHVILLE, UNITED STATES
David
Penneys
University of California, BERKELEY, UNITED STATES
Subfactors, planar algebras
We define a canonical planar *-algebra from a strongly Markov inclusion of finite von Neumann algebras. In the case of a connected unital inclusion of finite dimensional C*-algebras with the Markov trace, we show this planar algebra is isomorphic to the bipartite graph planar algebra of the Bratteli diagram of the inclusion. Finally, we show that a finite depth subfactor planar algebra is a planar subalgebra of the bipartite graph planar algebra of its principal graph.
Functional analysis
Category theory; homological algebra
Manifolds and cell complexes
General
301
337
10.4171/QT/23
http://www.ems-ph.org/doi/10.4171/QT/23
4
Fusion categories in terms of graphs and relations
Hendryk
Pfeiffer
University of British Columbia, VANCOUVER, CANADA
Fusion category, braided monoidal category, weak Hopf algebra, Tannaka–Kreĭn reconstruction, quiver
Every fusion category $\mathcal{C}$ that is $k$-linear over a suitable field $k$ is the category of finite-dimensional comodules of a weak Hopf algebra $H$. This weak Hopf algebra is finite-dimensional, cosemisimple and has commutative bases. It arises as the universal coend with respect to the long canonical functor $\omega\colon\mathcal{C}\to \mathbf{Vect}_k$. We show that $H$ is a quotient $H=H[\mathcal{G}]/I$ of a weak bialgebra $H[\mathcal{G}]$ which has a combinatorial description in terms of a finite directed graph $\mathcal{G}$ that depends on the choice of a generator $M$ of $\mathcal{C}$ and on the fusion coefficients of $\mathcal{C}$. The algebra underlying $H[\mathcal{G}]$ is the path algebra of the quiver $\mathcal{G}\times\mathcal{G}$, and so the composability of paths in $\mathcal{G}$ parameterizes the truncation of the tensor product of $\mathcal{C}$. The ideal $I$ is generated by two types of relations. The first type enforces that the tensor powers of the generator $M$ have the appropriate endomorphism algebras, thus providing a Schur–Weyl dual description of $\mathcal{C}$. If $\mathcal{C}$ is braided, this includes relations of the form ‘$RTT=TTR$’ where $R$ contains the coefficients of the braiding on $\omega M\otimes\omega M$, a generalization of the construction of Faddeev–Reshetikhin–Takhtajan to weak bialgebras. The second type of relations removes a suitable set of group-like elements in order to make the category of finite-dimensional comodules equivalent to $\mathcal{C}$ over all tensor powers of the generator $M$. As examples, we treat the modular categories associated with $U_q(\mathfrak{sl}_2)$.
Associative rings and algebras
Category theory; homological algebra
General
339
379
10.4171/QT/24
http://www.ems-ph.org/doi/10.4171/QT/24
Heegaard Floer homology as morphism spaces
Robert
Lipshitz
Columbia University, NEW YORK, UNITED STATES
Peter
Ozsváth
Princeton University, PRINCETON, UNITED STATES
Dylan
Thurston
Indiana University, BLOOMINGTON, UNITED STATES
Heegaard Floer homology, topological quantum field theories, bordered Floer homology, Hochschild homology
In this paper we prove another pairing theorem for bordered Floer homology. Unlike the original pairing theorem, this one is stated in terms of homomorphisms, not tensor products. The present formulation is closer in spirit to the usual TQFT framework, and allows a more direct comparison with Fukaya-categorical constructions. The result also leads to various dualities in bordered Floer homology.
Convex and discrete geometry
Differential geometry
General
381
449
10.4171/QT/25
http://www.ems-ph.org/doi/10.4171/QT/25