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The Jones polynomial and the Kauffman bracket are constructed, and their relation with knot and link theory is described. The quantum groups and tangle functor formalisms for understanding these invariants and their descendents are given.…

We develop the theory of probabilistic variants of the one-category and diagonal topological complexity, which bound the classical LS-category and topological complexity from below. Unlike any other classical or probabilistic invariants,…

This article is a continuation of work on construction and calculation various of modifications of invariant based on the use Euclidean metric values attributed to elements of manifold triangulation. We again address the well investigated…

Any four-dimensional Supersymmetric Quantum Field Theory with eight supercharges can be associated to a monoidal category of BPS line defects. Any Coulomb vacuum of such a theory can be conjecturally associated to an ``algebra of BPS…

In this paper, we introduce the concept of 3-alterfolds with embedded separating surfaces. When the separating surface is decorated by a spherical fusion category, we obtain quantum invariants of 3-alterfold, which is consistent with many…

In the generalized topological quantum field theory constructed by Andersen and Kashaev, invariants of 3-manifolds are defined given the combinatorial structure of a tetrahedral decomposition. Furthermore, a variant of the volume conjecture…

This is the first in a series of papers where we will derive invariants of three-manifolds and framed knots in them from the geometry of a manifold pseudotriangulation put in some way in a four-dimensional Euclidean space. Thus, the…

We describe a cohomological framework for measurement based quantum computation, in which symmetry plays a central role. Therein, the essential information about the computational output is contained in topological invariants, namely…

I discuss the relation between harmonic polynomials and invariant theory and show that homogeneous, harmonic polynomials correspond to ternary forms that are apolar to a base conic (the absolute). The calculation of Schlesinger that…

Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or…

There are exactly three finite subgroups of SU(2) that act irreducibly in the spin 1 representation, namely the binary tetrahedral, binary octahedral and binary icosahedral groups. In previous papers I have shown how the binary tetrahedral…

We show that when $r \geq 5$ is prime, the SO(3) Witten-Reshetikhin-Turaev quantum invariants for three-manifolds at the level $r$ form a dense set in the complex plane. This confirms a conjecture of Larsen and Wang.

The modular data of a modular category $\mathcal{C}$, consisting of the $S$-matrix and the $T$-matrix, is known to be an incomplete invariant of $\mathcal{C}$. More generally, the invariants of framed links and knots defined by a modular…

A braided monoidal category may be considered a $3$-category with one object and one $1$-morphism. In this paper, we show that, more generally, $3$-categories with one object and $1$-morphisms given by elements of a group $G$ correspond to…

Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic…

In the spirit of arXiv:2501.12985, we propose an abelian categorification of $\hat{Z}$-invariants for negative definite plumbed 3-manifolds. It provides a blueprint for the expected dictionary between these $3$-manifolds and log VOAs; that…

We review "quantum" invariants of closed oriented 3-dimensional manifolds arising from operator algebras.

We show that momentum-space tensor monopoles corresponding to nontrivial vector bundle generalizations, known as bundle gerbes, can be realized in bands of three-dimensional topological matter with nontrivial Hopf invariants. We provide a…

We introduce an equivariant version of contextuality with respect to a symmetry group, which comes with natural applications to quantum theory. In the equivariant setting, we construct cohomology classes that can detect contextuality. This…

First and second fundamental theorems are given for polynomial invariants of a class of pseudo-reflection groups (including the Weyl groups of type $B_n$), under the assumption that the order of the group is invertible in the base field.…