Explains connections between monopoles and modules on elliptic curves.
problem Understanding relationships between different mathematical objects.
method Explains equivalences between monopoles and polystable bundles and modules.
result Monopoles and polystable difference modules on elliptic curves are equivalent.
Study of Alexander modules for hyperplane arrangements, distinguishing complements.
problem Distinguishing homotopy equivalent but non-homeomorphic hyperplane arrangement complements.
method Analysis of twisted Alexander modules and polynomials.
result Distinguish non-homeomorphic homotopy equivalent arrangement complements.
Study fractional structures on bundle gerbe modules using rational homotopy theory.
problem Understanding twisted Chern classes of torsion bundle gerbe modules.
method Sullivan's rational homotopy theory to realize twisted Chern classes at the level of classifying spaces.
result Introduction of fractional U-structures as a universal framework.
Formula for Dehn twists in skein algebras defined on surfaces.
problem Calculating the effect of Dehn twists on skein algebras.
method Introduced filtrations of Kauffman bracket skein algebra and modules.
result Explicit formula for Dehn twists action.
The paper extends T-duality and Jacobi forms to Witten gerbe modules.
problem Extending T-duality and Jacobi forms to Witten gerbe modules.
method Constructing graded Hori maps and showing their isomorphisms on T-dual circle bundles, and constructing Witten gerbe modules.
result Graded twisted Chern characters of Witten gerbe modules are Jacobi forms under certain conditions.
Geometrically proves twisted Poincaré duality for orientable Poisson manifolds.
problem Establishing twisted Poincaré duality for Poisson manifolds.
method Geometrically reinterprets algebraic constructions of twisted Poisson modules and Poisson chain complexes.
result Explicit chain isomorphism between Poisson cochain and chain complexes with coefficients in Poisson modules.
Develops graphical calculus for monoidal categories with twisted pivotal structures.
problem Constructing modules for surfaces with Morse functions or foliations.
method Graphical calculus and string nets for monoidal categories with twisted pivotal structures.
result Twisted string net modules assemble in an oriented categorified 2-TQFT.
We define twisted Alexander polynomials of a complex hypersurface with arbitrary singularities. These generalize the classical Alexander polynomials of high dimensional hypersurfaces and the twisted Alexander polynomial of plane curves. We recover the classical torsionness and divisibility results, which say that, unde…
Dirac operator invertibility proven for specific manifolds.
problem Invertibility of twisted Dirac operator on manifolds.
method Closed connected spin manifold with non-negative scalar curvature, flat Hilbert module bundle.
result Dirac operator is invertible under given conditions.
Formula for Dehn twists on HOMFLY-PT skein modules with applications.
problem Understanding the action of Dehn twists on HOMFLY-PT skein modules.
method Introduced a formula for the action of Dehn twists on the HOMFLY-PT type skein module of a surface.
result Constructed an invariant \( z(M) \) for integral homology 3-spheres, finite type invariant of order \( n \).
We introduce an algebra Z[X,S] associated to a pair (X,S) of a virtual birack X and X-shadow S. We use modules over Z[X,S] to define enhancements of the virtual birack shadow counting invariant, extending the birack shadow module invariants to virtual case. We repeat this construction for the twisted virtual case. As a…
New algebraic structure derived from Hopf algebra and Drinfel'd twist.
problem Developing a new algebraic structure from existing mathematical concepts.
method Extending L∞-algebra to a Hopf algebra, twisting with Drinfel'd twist, and identifying Hopf morphisms and braided morphisms. result Braided L∞-algebra is derived from the process. We compute the Kauffman bracket skein module of the complement of a twist knot, finding that it is free and infinite dimensional. The basis consists of cables of a two-component link, one component of which is a meridian of the knot. The cabling of the meridian can be arbitrarily large while the cabling of the other co…
Study L-functions for knot group deformations, proving torsion and zero simplicity.
problem Investigate L-functions for knot group deformations.
method Analyze twisted knot modules and associated L-functions.
result Show torsion property and verify zero simplicity of L-functions.
Constructs differential models for twisted Spin^c-bordism and its dual, defining a new anomaly map.
problem Modeling and understanding twisted Spin^c-bordism and its dual.
method Geometric construction using bundle gerbes, gerbe modules, and eta-invariants.
result Definition of a twisted anomaly map from differential twisted K-theory to differential Anderson dual of twisted Spin^c-bordism.
Modified proof for a broader class of links.
problem Recovering twist number and volume bounds.
method Modified proof for a broader class of links.
result Modified proof works for a broader class of links.
We derive a general obstruction to the existence of Riemannian metrics of positive scalar curvature on closed spin manifolds in terms of hypersurfaces of codimension two. The proof is based on coarse index theory for Dirac operators that are twisted with Hilbert C*-module bundles. Along the way we give a complete and s…
It follows from earlier work of Silver-Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopf link. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexand…
Defines and calculates signature invariants for twisted linking forms.
problem Studying twisted linking forms of knots and three-manifolds.
method Describes how to define and calculate signature invariants associated to a linking form MimesMoF(t)/F[t±1] for F=R,C, where M is a torsion F[t±1]-module. result Classifies such linking forms up to isometry and Witt equivalence and studies their representability by matrices.
We prove, under some mild conditions, that the equivariant twisted K-theory group of a crossed module admits a ring structure if the twisting 2-cocycle is 2-multiplicative. We also give an explicit construction of the transgression map T1:H∗(Γ;A)→H∗−1((N⋊Γ;A) for any crossed module N→Γ and prove…
Study the algebraic action of torus on knot complement's skein module.
problem Understand the algebraic structure of knot complements and boundary tori.
method Analyze the Kauffman bracket skein algebra and module of the 3-twist knot complement.
result Determine the action of Kauffman bracket skein algebra on module of 3-twist knot complement.
Diagrammatic method calculates knot pairings in 3-sphere.
problem Computing knot pairings in 3-sphere.
method Diagrammatic computation of bilinear forms.
result Constructs bilinear forms on twisted Alexander modules.
This paper studies Jones-Wenzl idempotents in the twisted I-bundle over the Möbius band.
problem Understanding Jones-Wenzl idempotents in the Kauffman bracket skein module of the twisted I-bundle over the Möbius band.
method Analyzing the trace of Jones-Wenzl idempotents in the Kauffman bracket skein module of the twisted I-bundle over the Möbius band.
result Uncovering analog properties of Jones-Wenzl idempotents in the Kauffman bracket skein module of the twisted I-bundle over the Möbius band that differ from the annulus case.
Explicit matrix presentations of Blanchfield pairings and twisted pairings for torus knots.
problem Computing explicit matrix presentations of Blanchfield and twisted Blanchfield pairings for torus knots.
method Using a taut identity to construct a chain complex with few generators, and describing the twisted Alexander module.
result Explicit matrix presentations of the Blanchfield pairing and twisted pairings for (m,n)-torus knots. New formulas for coarse indices of twisted operators introduced.
problem Computing coarse indices of twisted Dirac type operators.
method Introduction of formulas in E-theory and K-theory.
result Index theoretic interpretation of duality between Roe algebra and stable Higson corona.
We exhibit a Poisson module restoring a twisted Poincare duality between Poisson homology and cohomology for the polynomial algebra R=C[X_1,...,X_n] endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for …
Real bundle gerbes classify geometric cycles in twisted KR-homology.
problem Classifying geometric cycles in twisted KR-homology.
method Real bundle gerbes and their modules as building blocks.
result Geometric cycles generate a real-oriented generalised homology theory.
Classifies differential operators on symplectic spinors in contact projective geometry.
problem Classifying differential operators on symplectic spinors.
method Classification of homomorphisms of generalized Verma modules and equivariant differential operators.
result Complete classification and construction of differential operators.
The paper studies power subgroups of Dehn twists in hyperelliptic mapping class groups.
problem Investigating the index of power subgroups in mapping class groups and hyperelliptic mapping class groups.
method Using a projective representation of mapping class groups through the Kauffman bracket skein module.
result The normal closure of the fifth power of a half-twist has infinite index in the mapping class group of a 2n-punctured sphere.
New findings show the Gilmer-Masbaum map isn't always one-to-one.
problem Determining the injectivity of the Gilmer-Masbaum map on Kauffman bracket skein modules.
method Computed the image of the evaluation map for specific cases of mapping tori and analyzed homology classes.
result The restriction of the Gilmer-Masbaum map to certain homology classes is not injective.
Given a bundle gerbe on a compact smooth manifold or, more generally, on a compact étale Lie groupoid M, we show that the corresponding category of gerbe modules, if it is non-trivial, is equivalent to the category of finitely generated projective modules over an Azumaya algebra on M. This result can be seen as an …
We introduce a Hilbert A-module structure on the higher oscillatory module, where A denotes the C∗-algebra of bounded endomorphisms of the basic oscillatory module. We also define the notion of an exterior covariant derivative in an A-Hilbert bundle and use it for a construction of an A-elliptic complex of d…
Study Kauffman module at -1+ε, conjectures relation to Reidemeister torsion.
problem Understanding Kauffman module and its relation to Reidemeister torsion.
method Introduced twisted self-linking, proved Kauffman relations at first order.
result Explicit relation between Kauffman module and Reidemeister torsion proved in particular cases.
The twisted torsion of a 3-manifold is well-known to be zero whenever the corresponding twisted Alexander module is non-torsion. Under mild extra assumptions we introduce a new twisted torsion invariant which is always non-zero. We show how this torsion invariant relates to the twisted intersection form of a bounding 4…
Introduces θ-almost twisted Poisson structures and their cohomology.
problem Characterizing and understanding θ-almost twisted Poisson structures. method Definition and construction of θ-almost twisted Poisson structures, Lie-Rinehart algebra, cochain complex, and cohomology. result Definition and construction of θ-almost twisted Poisson cohomology. Twisted spectral triples are a twisting of the notion of spectral triple aiming at dealing with some type III geometric situations. In the first part of the paper, we give a geometric construction of the index map of a twisted spectral triple in terms of σ-connections on finitely generated projective modules. This ma…
Researchers calculate dimensions of skein modules for 2-torus mapping tori.
problem Determining dimensions of Kauffman bracket skein modules for specific cases.
method Using generic q and decomposing twisted Hochschild homology of G-skein algebras. result Dimensions of skein modules for G=SL2 and G=GL1 are calculated. We establish the Thom isomorphism in twisted K-theory for any real vector bundle and develop the push-forward map in twisted K-theory for any differentiable proper map f:X→Y (not necessarily K-oriented). The push-forward map generalizes the push-forward map in ordinary K-theory for any K-oriented differentiable…
We construct bundles of modules of vertex operator algebras, and prove the rigidity and vanishing theorem for the Dirac operator on loop space twisted by such bundles. This result generalizes many previous results.
For a cyclic group A and a connected Lie group G with an A-module structure (with the additional conditions that G is compact and the A-module structure on G is 1-semisimple if $A\cong\ZZ$), we define the twisted Weyl group W=W(G,A,T), which acts on T and H1(A,T), where T is a maximal compact torus…
The Pontryagin dual of the twisted Alexander module for a d-component link and GL(N,Z) representation is an algebraic dynamical system with an elementary description in terms of colorings of a diagram. In the case of a knot, its associated topological entropy is the logarithmic growth rate of the number of torsion elem…
Research shows Khovanov homology can identify split links.
problem Detecting split links in knot theory.
method Applied Khovanov and Heegaard Floer homology theories.
result Khovanov homology detects split links.
New modules derived from Khovanov homology for links.
problem Constructing new link homology modules from Khovanov homology.
method Functoriality proof for cobordisms in 4D relative 1-handlebody complements.
result Functoriality of Rozansky-Willis's homology for cobordisms.
Diagrams and Reidemeister moves for links in a twisted S^1-bundle over an unorientable surface are introduced. Using these diagrams, we compute the Kauffman Bracket Skein Module (KBSM) of the connected sum of two projective spaces. In particular, we show that it has torsion. We also present a new computation of the KBS…
Develops Morse homology with DG coefficients for manifolds and spaces.
problem Homology with DG coefficients for manifolds and spaces.
method Derived local systems, DG modules, twisting cocycles, Morse trajectories.
result Isomorphic to DG Tor and Ext functors, recovers homology of total space of fibrations.
The abstract discusses homological stability in topological moduli spaces.
problem Homological stability in topological moduli spaces.
method Constructing canonical resolutions and introducing coefficient systems.
result Homological stability for various moduli spaces.
Study spherical twists on K3 surfaces, compute their centers.
problem Understanding autoequivalence groups of K3 surfaces.
method Introduced spherical twists, studied their intersection numbers, and classified subgroups.
result Computed the center of autoequivalence groups of K3 surfaces.
We introduce a framework, twisted parametrized stable homotopy theory, for describing semi-infinite homotopy types. A twisted parametrized spectrum is a section of a bundle whose fibre is the category of spectra. We define these bundles in terms of modules over a stack of parametrized spectra and in terms of diagrams o…