A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Skein theory classifies UFCs with specific fusion rules.
We consider two approaches to isotopy invariants of oriented links: one from ribbon categories and the other from generalized Yang-Baxter operators with appropriate enhancements. The generalized Yang-Baxter operators we consider are obtained from so-called gYBE objects following a procedure of Kitaev and Wang. We show …
In a previous paper, we introduced special types of fusions, so called simple-ribbon fusions on links. A knot obtained from the trivial knot by a finite sequence of simple-ribbon fusions is called a simple-ribbon knot. Every ribbon knot with <10 crossings is a simple-ribbon knot. In this paper, we give a formula for th…
New insights into knot fusion numbers via cabling.
We construct a state-sum type invariant of smooth closed oriented -manifolds out of a -crossed braided spherical fusion category (-BSFC) for a finite group. The construction can be extended to obtain a -dimensional topological quantum field theory (TQFT). The invariant of -manifolds generalizes s…
Defines state sum models with defects in 3-manifolds.
Study knots that divide ribbon knotted surfaces, computing their half ribbon genus and fusion number.
The abstract introduces a method to compute surface invariants in 4-manifolds.
Revised proof shows ribbonness of surface-links in 4-sphere.
Semisimple 4D field theories can't distinguish smooth 4-manifolds.
We introduce semisimple 2-categories, fusion 2-categories, and spherical fusion 2-categories. For each spherical fusion 2-category, we construct a state-sum invariant of oriented singular piecewise-linear 4-manifolds.
New 4-manifold invariant defined from trisection diagrams.
The paper defines cyclic sets from ribbon string links and connects them to quantum invariants.
We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To facilitate the discussion we describe a convenient, concrete and useful variation o…
The Reshetikhin-Turaev invariant, Turaev's TQFT, and many related constructions rely on the encoding of certain tangles (n-string links, or ribbon n-handles) as n-forms on the coend of a ribbon category. We introduce the monoidal category of Hopf diagrams, and describe a universal encoding of ribbon string links as Hop…
We define a symmetric monoidal (4,3)-category with duals whose objects are certain enriched multi-fusion categories. For every modular tensor category , there is a self enriched multi-fusion category giving rise to an object of this symmetric monoidal (4,3)-category. We conjecture that the e…
Quantum invariant derived from ternary cohomology of self-distributive structures.
Functor connects 4D 2-handlebodies to ribbon categories, detecting non-deformation diffeomorphisms.
New skein categories for non-semisimple settings, extending existing theory.
For a group G, the notion of a ribbon G-category was introduced by the second author in a previous work with a view towards constructing 3-dimensional homotopy quantum field theories (HQFT's) with target K(G,1). We discuss here how to derive ribbon G-categories from a simple complex Lie algebra g where G is the center …
New (3+1) TQFTs created from non-semisimple categories.
We define the notion of a Kirby element of a ribbon category C (not necessarily semisimple). Kirby elements lead to 3-manifolds invariants. We characterize (in terms of the structure maps of some categorical Hopf algebra) a set of Kirby elements of C which is sufficiently large to recover the known quantum invariants c…
Given a connected cobordism between two knots in the 3-sphere, our main result is an inequality involving torsion orders of the knot Floer homology of the knots, and the number of local maxima and the genus of the cobordism. This has several topological applications: The torsion order gives lower bounds on the bridge i…
We construct what we call a Kirby category, a monoidal category whose morphisms are smooth 4-manifolds, projecting down to another monoidal category whose morphisms are orientable 3-manifolds, the projection being induced by the boundary map on manifolds. We construct a higher categorical generalization of such concept…
3D HQFTs constructed using graded monoidal categories.
Internalizes Turaev's construction for TQFTs using ribbon categories.
In this paper, we extend the notion of modular functor and fusion category to what we called equivariant modular functor and equivariant fusion category, where is a finite group, and establish a correspondence between between these notions.
New categories from TQFTs interpret skein relations.
We define a category of tangles diagrams drawn on surfaces with boundaries. On the one hand we show that there is a natural functor from the category of virtual tangles to which induces an equivalence of categories. On the other hand, we show that is universal among ribbon c…
Calculates Dehn twist actions on conformal blocks for modular categories.
A new approach to Morse theory using folded ribbon trees.
New evidence refutes old conjectures about knot homology ranks, suggesting new congruences.
3D HQFT comparison proves state sum equals surgery.
We present an invariant of connected and oriented closed 3-manifolds based on a coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant can be understood as the generalization to Weak Hopf Algebras of the Hennings-Kauffman-Radford evaluation of an unoriented framed link using a dual quantum-trace. Thi…
Involutive Hopf monoids yield surface invariants.
Quantum invariants for surfaces in 4D 2-handlebodies.
A string-net model associates a vector space to a surface in terms of graphs decorated by objects and morphisms of a pivotal fusion category modulo local relations. String-net models are usually considered for spherical fusion categories, and in this case the vector spaces agree with the state spaces of the correspondi…
Defines slice depth for 2-knots and sets upper bounds for specific knots.
Study on prime knots, slice obstructions, and ribbon concordances.
Turaev-Viro invariants match for certain surface bundles.
We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations…
We prove that the category of abelian gerbes with connection over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These bundles are equipped with a connection and with a "fusion" product with respect to triples of paths. The equivalence is established by explicit fun…
Propose a model-independent axiomatic framework for derived skein theory.
The paper classifies homotopy ribbon discs with specific groups.
A family of TQFTs parametrised by G-crossed braided spherical fusion categories has been defined recently as a state sum model and as a Hamiltonian lattice model. Concrete calculations of the resulting manifold invariants are scarce because of the combinatorial complexity of triangulations, if nothing else. Handle deco…
The paper constructs braiding structures for a specific subfactor.
It is known that every ribbon category with unimodality allows symmetrized -symbols with full tetrahedral symmetries while a spherical category does not in general. We give an explicit counterexample for this, namely the category . We define the mirror conjugate symmetry of -symbols instead and sho…
Given a rational homology sphere which bounds rational homology balls, we investigate the complexity of these balls as measured by the number of 1-handles in a handle decomposition. We use Casson-Gordon invariants to obtain lower bounds which also lead to lower bounds on the fusion number of ribbon knots. We use Levine…