Defines a universal state sum construction for various TQFTs.
problem No specific problem stated; universal construction for TQFTs.
method Defines a universal state sum construction using n-categories with specific conditions.
result Produces state sums from n-categories and handle decompositions of n+1-manifolds.
This research extends TQFTs to include defects using state-sum construction.
problem Extending topological quantum field theories to include defects.
method General state-sum construction with extended Pachner moves, requiring flag-likeness of triangulations.
result Derivation of equations and string diagrams for 2D TQFTs with defects.
Innovative 2-categories create 4-manifold invariants.
problem Constructing invariants for 4-manifolds.
method Semisimple 2-categories, fusion 2-categories, and state-sum construction.
result Construct a state-sum invariant for 4-manifolds.
The paper characterizes boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
problem Characterizing boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
method Identifying explicit boundary locality conditions and proving consistency with state sum models.
result Turaev-Viro and Dijkgraaf-Witten theories with boundary defects admit a state sum description.
3D HQFTs constructed using graded monoidal categories.
problem Constructing 3D HQFTs with specific targets.
method Using spherical χ-fusion categories and the state sum method.
result 3D HQFTs constructed with target Bχ.
New invariant for spin 3-manifolds using super 3-cocycles.
problem Constructing an invariant for spin 3-manifolds.
method State sum construction based on super 3-cocycles and combinatorial representation.
result The invariant is sensitive to the spin structure.
Constructs a path integral for fermionic SPTs, solving anomalies in 2+1D topological orders.
problem Anomalies in (2+1)D fermionic topological phases and their computation.
method Combining (2+1)D fermionic topological order with symmetry fractionalization data to construct a (3+1)D path integral.
result Reproduces the Z16 anomaly indicator for time-reversal symmetric topological superconductors. The paper constructs quantum invariants for knotoid diagrams.
problem Quantum invariants for knotoid diagrams in R2. method Decompose Morse knotoid diagrams into basic elementary diagrams, each associated with a matrix solving the quantum Yang-Baxter equation. Define quantum state sum models to recover various polynomials.
result Recover and define new polynomials for Morse knotoids.
State sums for quantum link invariants from a specific representation.
problem Calculating quantum link invariants from a specific representation of U_q(gl_{N|M}).
method Using state sums and representation theory of U_q(gl_{N|M}).
result Explicit relation with Kashaev invariants for the N-th exterior power of the standard representation.
Defines state sum models with defects in 3-manifolds.
problem Detecting and characterizing defects in 3-manifolds.
method Turaev-Viro-Barrett-Westbury state sum models with defects labeled by bimodule categories and functors.
result State sums are triangulation-independent and can be computed using polygon diagrams.
New invariant valued in pictures enhances state-sum invariants.
problem Enhancing state-sum invariants for virtual knots.
method Constructing a picture-valued biquandle bracket.
result Enhances various state-sum invariants for virtual knots.
Constructs a new invariant for 4-manifolds and 3+1 TQFTs.
problem Developing a new invariant for 4-manifolds and 3+1 TQFTs.
method Using a G-crossed braided spherical fusion category to construct a state-sum type invariant.
result Introduces a new invariant for 4-manifolds and 3+1 TQFTs.
In this paper, we calculate the values of the E6 state sum invariants for the lens spaces L(p,q). In particular, we show that the values of the invariants are determined by pmod12 and qmod(p,12). As a corollary, we show that the E6 state sum is a homotopy invariant for the oriented lens spaces.
3D HQFT comparison proves state sum equals surgery.
problem Comparing two HQFT approaches for 3D.
method Proving isomorphism between state sum and surgery methods.
result State sum 3D HQFT isomorphic to surgery 3D HQFT.
In this paper, we provide a construction of a state-sum model for finite gauge-group Dijkgraaf-Witten theory on surfaces with codimension 1 defects. The construction requires not only that the triangulation be subordinate to the filtration, but flag-like: each simplex of the triangulation is either disjoint from the de…
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…
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.
New construction of Turaev-Viro invariants invariant under Morita equivalence.
problem Constructing Turaev-Viro invariants invariant under Morita equivalence.
method Pivotal bicategory construction of spherical module categories.
result The invariant recovers the standard Turaev-Viro invariant and is independent of the skeleton.
Given a discrete group G and a spherical G-fusion category whose neutral component has invertible dimension, we use the state-sum method to construct a 3-dimensional Homotopy Quantum Field Theory (HQFT) with target the Eilenberg-MacLane space K(G,1).
This paper is a survey of several papers in quandle homology theory and cocycle knot invariants that have been published recently. Here we describe cocycle knot invariants that are defined in a state-sum form, quandle homology, and methods of constructing non-trivial cohomology classes.
Defines a new link invariant for type D webs.
problem Invariants for framed unoriented links in type D.
method Positive state sum for type D webs, derived from quantum so(2N) modules.
result Relates to Reshetikhin-Turaev's invariant.
Algorithm calculates quantum invariants of 3-manifolds with polynomial time complexity.
problem Computing quantum invariants from Tambara-Yamagami categories is #P-hard.
method Fixed-parameter tractable algorithm with first Betti number as parameter.
result Existence of FPT algorithm for Tambara-Yamagami invariants.
New proof for knot state-sum formula using bijection between states.
problem Proving a knot state-sum formula for colored Jones polynomial.
method Established bijection between states on arc-graph and bichromatic digraph, used flow property of R-matrix.
result Two state models are essentially the same, extending formula to links.
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 work in the reduced SU(N,K) modular category as constructed recently by Blanchet. We define spin type and cohomological refinements of the Turaev-Viro invariants of closed oriented 3-manifolds and give a formula relating them to Blanchet's invariants. Roberts' definition of the Turaev-Viro state sum is exploited. Fu…
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…
Paper constructs a HOMFLYPT-type invariant for pseudo links.
problem Inability to construct polynomial invariants for pseudo links using Hecke algebra techniques.
method Using a resolution homomorphism and pseudo Hecke algebra of type \(A\), the paper constructs a HOMFLYPT-type invariant for oriented pseudo links.
result The constructed invariant satisfies a natural pseudo skein relation and admits a state-sum formulation.
Homology and cohomology theory for topological quandles computed.
problem Computing invariants for knot diagrams using quandle cocycles.
method Introducing homology and cohomology theory for topological quandles, studying their relation to quandle groups, and using topological quandle cocycles to compute state sum invariants.
result State sum invariants computed using topological quandle cocycles.
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{…
Paper describes a state sum formula for a graph coloring polynomial.
problem Counting n-face colorings of ribbon graphs for various n. method Combines topological quantum field theory and diagrammatic tensors.
result Describes a state sum formula for the total face color polynomial.
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…
The paper defines new polynomials for links and linkoids.
problem No specific problem stated; focus on new polynomials.
method Defined as sums over states of link or linkoid diagrams with f=n. result Constructed new polynomials for starred links and linkoids.
New method uses quandle rings to distinguish knots and their mirrors.
problem Distinguishing knots and their mirror images.
method Combining idempotents in quandle rings with state sum invariants.
result Improved knot and mirror image distinction using fewer quandles.
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…
Paper derives explicit formulas for AJ-bracket of tied links.
problem Lack of state-sum formula for AJ-bracket of tied links.
method Analyzed AJ-states of 2- and 3-tied link diagrams, derived resolution trees, and state-sum formulas.
result Derives first closed-form expressions for AJ-bracket.
Paper discusses biquandle cohomology and invariants for surface-links.
problem Computing invariants for surface-links using biquandle theory.
method Developed (co)homology theory of biquandles, introduced new invariants using broken surface diagrams and marked graph diagrams.
result Constructs new invariants for surface-links using biquandle theory.
New invariants for RNA foldings and stuck links defined.
problem Defining invariants for RNA foldings and stuck links.
method Assigning Boltzmann weights at classical and stuck crossings.
result Explicit computations of new invariants provided.
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…
New method calculates manifold invariants using Kirby diagrams with 3-handles.
problem Calculating manifold invariants from TQFTs is complex due to combinatorial complexity.
method Reformulated state sum model using Kirby diagrams with 3-handles and graphical calculus.
result Invariants are multiplicative under connected sum, detecting no exotic structures.
New formulas connect knot invariants with theta functions.
problem Proving conjectures about knot invariants and 3-manifold invariants.
method Inverted state sums and Habiro series.
result Discovered formulas relating knot invariants to theta functions.
We provide a calculus for the presentation of closed 3-manifolds via nullhomotopic filling Dehn spheres and we use it to define an invariant of closed 3-manifolds by applying the state-sum machinery. As a potential application of this invariant, we show how to get lower bounds for the Matveev complexity of P2-irreducib…
Aicardi's invariant F(L) is extended to colored singular links using graphical calculus.
problem Constructing an invariant for colored classical and singular links.
method State-sum model using graphical calculus for oriented, colored, 4-valent planar graphs.
result Extends F(L) to colored singular links, showing it's stronger than HOMFLY-PT polynomial. New method distinguishes knots and knotted surfaces.
problem Distinguishing knots and knotted surfaces.
method Twisted set-theoretic Yang-Baxter solutions and Alexander numbering.
result Distinguished 2-twist spun trefoil from its reverse. Unified invariant for immersed surface-links using biquandle cocycles.
problem Invariants for immersed surface-links in 4-space.
method Biquandle cohomology, Roseman moves, singular biquandle 3-cocycles.
result Invariance of state-sum invariant under all generating moves, including singular move (h).
Paper uses Turaev-Viro TQFT to estimate 3-manifold genus.
problem Estimating the Heegaard genus of 3-manifolds.
method Turaev-Viro state sum TQFT and unitary modular category.
result Provides a lower bound for Heegaard genus using TQFT.
This paper has been withdrawn because there is a fundamental error in the computations; with the right computational scheme it seems to be just a version of the Jones polynomial
Homomorphisms on quandle cohomology groups that raise the dimensions by one are studied in relation to the cocycle state-sum invariants of knots and knotted surfaces. Skein relations are also studied.