arXiv research

Defines a universal state sum construction for various TQFTs.

Defines state sum models with defects in 3-manifolds.

New proof for knot state-sum formula using bijection between states.

In this paper, we calculate the values of the $E_6$ state sum invariants for the lens spaces $L(p,q)$. In particular, we show that the values of the invariants are determined by $p \mod 12$ and $q \mod (p,12)$. As a corollary, we show that the $E_6$ state sum is a homotopy invariant for the oriented lens spaces.

The paper characterizes boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.

In this paper we give a short introduction to our results on the holonomy of gerbe-connections and explain our motivation coming from state-sum models.

This work studies the problem of stochastic dynamic filtering and state propagation with complex beliefs. The main contribution is GP-SUM, a filtering algorithm tailored to dynamic systems and observation models expressed as Gaussian Processes (GP), and to states represented as a weighted sum of Gaussians. The key attr…

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.

Defines a new link invariant for type D webs.

Global EQG sums boundary states over manifold diffeomorphism classes.

We develop a diagrammatic formalism for calculating the Alexander polynomial of the closure of a braid as a state-sum. Our main tools are the Markov trace formulas for the HOMFLY-PT polynomial and Young's semi-normal representations of the Iwahori-Hecke algebras of type A.

We present state sums for quantum link invariants arising from the representation theory of $U_q(\mathfrak{gl}_{N|M})$. We investigate the case of the $N$-th exterior power of the standard representation of $U_q(\mathfrak{gl}_{N|1})$ and explicit the relation with Kashaev invariants.

We demonstrate the triangulability of compact 3-dimensional topological pseudomanifolds and study the properties of such triangulations, including the Hauptvermutung and relations by Alexander star moves and Pachner bistellar moves. We also provide an application to state-sum invariants of 3-dimensional topological pse…

SPTN uses invertible transformations to improve sum-product networks.

Homology and cohomology theory for topological quandles computed.

Paper derives explicit formulas for AJ-bracket of tied links.

We develop a dimer model for the Alexander polynomial of a knot. This recovers Kauffman's state sum model for the Alexander polynomial using the language of dimers. By providing some additional structure we are able to extend this model to give a state sum formula for the twisted Alexander polynomial of a knot dependin…

In this paper we define a new state sum based on the regions defined by tangles on a surface which is an oriented closed surface with a finite number of open holes drilled. From this state sum we obtain an invariant of regular isotopy for the tangles named $u$-invariant. The values of the $u$-invariant are in $\mathbb{…

In this paper, we characterize the sigma-adequacy of a link diagram in two ways: in terms of a certain edge subset of its Tait graph and in terms of a certain product of Tutte polynomials. Furthermore, we show that the symmetrized Tutte polynomial of the Tait graph of a link diagram can be written as a sum of these pro…

Paper describes a state sum formula for a graph coloring polynomial.

We derive the general state sum construction for 2D topological quantum field theories (TQFTs) with source defects on oriented curves, extending the state-sum construction from special symmetric Frobenius algebra for 2-D TQFTs without defects (cf. Lauda \& Pfeiffer \cite{LP}). From the extended Pachner moves (Crane \& …

Constructs a path integral for fermionic SPTs, solving anomalies in 2+1D topological orders.

The paper defines new polynomials for links and linkoids.

The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a state-sum invariant for classical knots and knotted surfaces is developed via a cohomology theory of racks a…

New formulas connect knot invariants with theta functions.

New invariant for spin 3-manifolds using super 3-cocycles.

Study shows specific states produce Khovanov homology torsion.

3D HQFTs constructed using graded monoidal categories.

New method uses quandle rings to distinguish knots and their mirrors.

We prove that if two Tambara-Yamagami categories TY(A,χ,ν) and TY(A',χ',ν') give rise to the same state sum invariants of 3-manifolds and the order of one of the groups A, A' is odd, then ν=ν' and there is a group isomorphism A\approx A' carrying χto χ'. The proof is based on an explicit computation of the state sum in…

We consider the problem of two-player zero-sum games. This problem is formulated as a min-max Markov game in the literature. The solution of this game, which is the min-max payoff, starting from a given state is called the min-max value of the state. In this work, we compute the solution of the two-player zero-sum game…

We study 2-string free tangle decompositions of knots with tunnel number two. As an application, we construct infinitely many counter-examples to a conjecture in the literature stating that the tunnel number of the connected sum of prime knots doesn't degenerate by more than one.

New algorithm finds near-optimal policies efficiently in zero-sum games.

A new method extracts features from time series data using iterated sums and improves classification accuracy.

This paper develops a Hoeffding inequality for the partial sums $\sum_{k=1}^n f (X_k)$, where $\{X_k\}_{k \in \mathbb{Z}_{> 0}}$ is an irreducible Markov chain on a finite state space $S$, and $f : S \to [a, b]$ is a real-valued function. Our bound is simple, general, since it only assumes irreducibility and finiteness…

New invariants for RNA foldings and stuck links defined.

This paper continues the study of periodic links started in \cite{Politarczyk2}. It contains a study of the equivariant analogues of the Jones polynomial, which can be obtained from the equivariant Khovanov homology. In this paper we describe basic properties of such polynomials, show that they satisfy an analogue of t…

Algorithm calculates quantum invariants of 3-manifolds with polynomial time complexity.

We produce a facial state sum on plane diagrams of a knot or a link which admits an invariant specialization under Polyak's recent set of generating of 4 Reidemeister moves. Thus an isotopy invariant of framed links is obtained. Each state is a complete coloring of the faces of the diagram into white and black faces so…

New method improves missing mass concentration bounds.

Optimizes non-linear outcomes from summed contributions.

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…

Let $N$ be a closed enlargeable manifold in the sense of Gromov-Lawson and $M$ a closed spin manifold of equal dimension, a famous theorem of Gromov-Lawson states that the connected sum $M\# N$ admits no metric of positive scalar curvature. We present a potential generalization of this result to the case where $M$ is n…

Crane and Frenkel proposed a state sum invariant for triangulated 4-manifolds.They defined and used new algebraic structures called Hopf categories for their construction. Crane and Yetter studied Hopf categories and gave some examples using group cocycles that are associated to the Drinfeld double of a finite group. I…

The paper constructs quantum invariants for knotoid diagrams.

Researchers found a quadratic estimate for embedding higher-dimensional simplices into sphere-connected sums.

Bayesian method synthesizes barrier certificates for unknown systems with latent states.

We prove a discrete Gauss-Bonnet-Chern theorem which states where summing the curvature over all vertices of a finite graph G=(V,E) gives the Euler characteristic of G.