Narrow Results

We use the theory of sutured TQFT to classify contact elements in the sutured Floer homology, with ℤ coefficients, of certain sutured manifolds of the form (Σ × S¹, F × S¹) where Σ is an annulus or punctured torus. Using this classification, we give a new proof that the contact invariant in sutured Floer homology with ℤ coefficients of a contact structure with Giroux torsion vanishes. We also give a new proof of Massot's theorem that the contact invariant vanishes for a contact structure on (Σ × S¹, F × S¹) described by an isolating dividing set.

We extend the topological field theory (itsy bitsy topological field theory) of our previous work from mod 2 to twisted coefficients. This topological field theory is derived from sutured Floer homology (SFH) but described purely in terms of surfaces with signed points on their boundary (occupied surfaces) and curves on those surfaces respecting signs (sutures). It has information-theoretic (itsy) and quantum-field-theoretic (bitsy) aspects. In the process we extend some results of SFH, consider associated ribbon graph structures, and construct explicit admissible Heegaard decompositions.

The purpose of this paper is to study some obstruction classes induced by a construction of a homotopy-theoretic version of projective TQFT (projective HTQFT for short). A projective HTQFT is given by a symmetric monoidal projective functor whose domain is the cospan category of pointed finite CW-spaces instead of a cobordism category. We construct a pair of projective HTQFT’s starting from a ${Hopf}_k^{bc}$-valued Brown functor where ${Hopf}_k^{bc}$ is the category of bicommutative Hopf algebras over a field $k$ : the cospanical path-integral and the spanical path-integral of the Brown functor. They induce obstruction classes by an analogue of the second cohomology class associated with projective representations. In this paper, we derive some formulae of those obstruction classes. We apply the formulae to prove that the dimension reduction of the cospanical and spanical path-integrals are lifted to HTQFT’s. In another application, we reproduce the Dijkgraaf–Witten TQFT and the Turaev–Viro TQFT from an ordinary ${Hopf}_k^{bc}$-valued homology theory.

In this paper, we study the variation of the Turaev–Viro invariants for $3$-manifolds with toroidal boundary under the operation of attaching a $(p,q)$-cable space. We apply our results to a conjecture of Chen and Yang which relates the asymptotics of the Turaev–Viro invariants to the simplicial volume of a compact oriented $3$-manifold. For $p$ and $q$ coprime, we show that the Chen–Yang volume conjecture is stable under $(p,q)$-cabling. We achieve our results by studying the linear operator ${\mathrm{\text{RT}}}_r$ associated to the torus knot cable spaces by the Reshetikhin–Turaev ${\mathrm{\text{SO}}}_3$-Topological Quantum Field Theory (TQFT), where the TQFT is well-known to be closely related to the desired Turaev–Viro invariants. In particular, our utilized method relies on the invertibility of the linear operator for which we provide necessary and sufficient conditions.

The notion of 2-framed three-manifolds is defined. The category of 2-framed cobordisms is described, and used to define a 2-framed three-dimensional TQFT. Using skeletonization and special features of this category, a small set of data and relations is given that suffice to construct a 2-framed three-dimensional TQFT. These data and relations are expressed in the language of surgery.

The Jones polynomial of a knot in 3-space is a Laurent polynomial in q, with integer coefficients. Many people have pondered why this is so, and what a proper generalization of the Jones polynomial for knots in other closed 3-manifolds is. Our paper centers around this question. After reviewing several existing definitions of the Jones polynomial, we argue that the Jones polynomial is really an analytic function, in the sense of Habiro. Using this, we extend the holonomicity properties of the colored Jones function of a knot in 3-space to the case of a knot in an integer homology sphere, and we formulate an analogue of the AJ Conjecture. Our main tools are various integrality properties of topological quantum field theory invariants of links in 3-manifolds, manifested in Habiro's work on the colored Jones function.

Apart from their generic relationship to knots and their application to particle physics [1], flattened Moebius strips (FMS) are of intrinsic interest as elements of a genus with specific rules of combination and a unique taxonomy. Here, FMS taxonomy is developed in detail from combinatorial and lexicographic points of view which include notions of degeneracy, completeness and excited states. The results are compared to the standard, spin-parameterized, abstract hierarchy derived by group-theoretic arguments as the direct product of vector spin spaces [2]. A review of the notion of excited states then leads to a new and different model of Beta decay that employs only fusion and fission. There is additional discussion of the relationship between twist and charge and an operator/tensor formulation of the fusion and fission of basic FMS units. Associating a Hopf algebra to FMS operations as a step toward a topological quantum field theory is also investigated. The notion of spinor/twistor networks is seen to emerge from a consideration of FMS configurations for higher values of twist and the introduction of a mode dual to the canonical FMS configuration. The last section discusses the connection of the MS genus to fiber bundle/gauge theory, the concept of spin, and the Dirac equation of the electron.

A knot diagram can be divided by a circle into two parts, such that each part can be coded by a planar tree with integer weights on its edges. A half of the number of intersection points of this circle with the knot diagram is called the girth. The girth of a knot is the minimal girth of all diagrams of this knot. The girth of a knot minus one is an upper bound of the Heegaard genus of the 2-fold branched covering of that knot. We will use Topological Quantum Field Theory (TQFT) coming from the Kauffman bracket to determine the girth of some knots. Consequently, our method can be used to determine the Heegaard genus of the 2-fold branched covering of some knots.

We give an alternative presentation of Khovanov homology of links. The original construction rests on the Kauffman bracket model for the Jones polynomial, and the generators for the complex are enhanced Kauffman states. Here we use an oriented sl(2) state model allowing a natural definition of the boundary operator as twisted action of morphisms belonging to a TQFT for trivalent graphs and surfaces. Functoriality in original Khovanov homology holds up to sign. Variants of Khovanov homology fixing functoriality were obtained by Clark–Morrison–Walker [7] and also by Caprau [6]. Our construction is similar to those variants. Here we work over integers, while the previous constructions were over gaussian integers.

We construct functors from a certain algebraic category , defined by Hopf algebra generators and relations, to the category of vector spaces, based on spherical categories. The category is proposed by Habiro to be isomorphic to the cobordism category of once-punctured surfaces. If the proposal is proved valid, the result of this paper would imply a construction of a TQFT functor based on a spherical category.

By methods similar to those used by L. Jeffrey [L. C. Jeffrey, Chern–Simons–Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Commun. Math. Phys.147 (1992) 563–604], we compute the quantum SU(N)-invariants for mapping tori of trace 2 homeomorphisms of a genus 1 surface when N = 2, 3 and discuss their asymptotics. In particular, we obtain directly a proof of a version of Witten's asymptotic expansion conjecture for these 3-manifolds. We further prove the growth rate conjecture for these 3-manifolds in the SU(2) case, where we also allow the 3-manifolds to contain certain knots. In this case we also discuss trace -2 homeomorphisms, obtaining — in combination with Jeffrey's results — a proof of the asymptotic expansion conjecture for all torus bundles.

We show that any twisted Dijkgraaf–Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image. This generalizes work of Etingof et al. showing that the braid group images are finite [P. Etingof, E. C. Rowell and S. Witherspoon, Braid group representations from twisted quantum doubles of finite groups, Pacific J. Math.234 (2008)(1) 33–42]. In particular, our result answers their question regarding finiteness of images of arbitrary mapping class group representations in the affirmative.

Our approach is to translate the problem into manipulation of colored graphs embedded in the given surface. To do this translation, we use the fact that any twisted Dijkgraaf–Witten representation associated to a finite group $G$ and 3-cocycle $\omega$ is isomorphic to a Turaev–Viro–Barrett–Westbury (TVBW) representation associated to the spherical fusion category ${\mathrm{\text{Vec}}}_G^{\omega }$ of twisted $G$-graded vector spaces. The representation space for this TVBW representation is canonically isomorphic to a vector space of ${\mathrm{\text{Vec}}}_G^{\omega }$-colored graphs embedded in the surface [A. Kirillov, String-net model of Turaev-Viro invariants, Preprint (2011), arXiv:1106.6033]. By analyzing the action of the Birman generators [J. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math.22 (1969) 213–242] on a finite spanning set of colored graphs, we find that the mapping class group acts by permutations on a slightly larger finite spanning set. This implies that the representation has finite image.

We provide a description of adequate categorical data to give a Turaev–Viro type state-sum construct of invariants of 3-manifolds with a system of defects, generalizing the Dijkgraaf–Witten type invariants of our earlier work. We term the defects in our construction defects-with-structure because algebraic data associated to them is in general richer than a module category over the spherical fusion category from which the theory is constructed when no defect is present.

We apply the theory of directed topology developed by Grandis [Directed homotopy theory, I. The fundamental category, Cah. Topol. Géom. Différ. Catég. 44 (2003) 281–316; Directed Algebraic Topology: Models of Non-Reversible Worlds, New Mathematical Monographs, Vol. 13 (Cambridge University Press, Cambridge, 2009)] to the study of stratified spaces by describing several ways in which a stratification or a stratification with orientations on the strata can be used to produce a related directed space structure. This description provides a setting for the constructions of state-sum TQFTs with defects of [A. L. Dougherty, H. Park and D. N. Yetter, On 2-dimensional Dikjgraaf-Witten theory with defects, J. Knot Theory Ramifications 25(5) (2016) 1650021, doi:10.1142/S0218216516500218; I. J. Lee and D. N. Yetter, Dijkgraaf–Witten type invariants of Seifert surfaces in 3-manifolds, J. Knot Theory Ramifications 26(5) (2017) 1750026, doi:10.1142/S0218216517500262], which we extend to a similar construction of a Dijkgraaf–Witten type TQFT in the case where the defects (lower-dimensional strata) are not sources or targets, but sources on one side and targets on the other, according to an orientation convention.

The aim of this note is to give two applications of Topological Quantum Field Theories to the topology of open manifolds. The invariants we derived may be used to test if an open manifold is simply connected at infinity (as we did for Whitehead's manifold in case of the sl₂(C)-TQFT in level 3) or to construct examples of non-homeomorphic contractible open 3-manifolds.

Let M be the manifold obtained by 0-framed surgery along a knot K in the 3-sphere. A Topological Quantum Field Theory assigns to a fundamental domain of the universal abelian cover of M an operator, whose non-nilpotent part is the Turaev-Viro module of K. In this paper, using surgery formulas, we give a matrix presentation for the Turaev-Viro module of any knot K, in the case of the (V_p, Z_p) TQFT of Blanchet, Habegger, Masbaum and Vogel. We do the computation for a family of knots in the special case p = 8, and note the relation with the fibering question.

We use the Topological Quantum Field Theory derived from the skein theory of the Kauffman bracket to compute TQFT-invariants at infinity for Whitehead's manifold.

We prove that the Kauffman bracket skein module of a 3-manifold M at a 4r-th root of unity divided by the Jones-Wenzl idempotent depends only on ∂M.

In this note we describe a representation theoretic approach to functorial functor valued knot invariants with the focus on (categorified) Schur-Weyl dualities. Applications include categorified Reshetikhin-Turaev invariants, an extension of Khovanov homology and a diagrammatical description of the category of finite dimensional GL(m|n)-modules.

By using quantum Teichmüller theory, a new type of three-dimensional TQFT has been constructed with the following distinguishing features: it uses the combinatorial framework of shaped triangulations; it takes its values in the space of tempered distributions; the fundamental building block of the theory is given by Faddeev's quantum dilogarithm. The semi-classical behavior and the geometrical ingredients suggest that the constructed TQFT is related to exact solution of quantum Chern–Simons theory with gauge group SL(2, ℂ). We also remark that quantum Teichmüller theory itself admits an additional real parameter which preserves unitarity but affects the projective factor in the corresponding mapping class group representation.