arXiv research

The paper classifies PD_4-complexes based on their fundamental group properties.

The goal of this work is to study the ideals of the Goldman Lie algebra $S$. To do so, we construct an algebra homomorphism from $S$ to a simpler algebraic structure, and focus on finding ideals of this new structure instead. The structure $S$ can be regarded as either a $\mathbb{Q}$-module or a $\mathbb{Q}$-module gen…

A commuting $n$-tuple $(T_1, \ldots, T_n)$ of bounded linear operators on a Hilbert space $\clh$ associate a Hilbert module $\mathcal{H}$ over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \rightarrow \mathcal{H}, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h…

We show that every $\mathbb{Z}$-torsion free knot module is realized by a ribbon 2-knot with group of geometric dimension at most 2, and give some partial results on the characterization of the knot modules of fibred ribbon 2-knots.

The study finds stably free modules and distinct 2-complexes for large ranks.

In this paper, we consider the similarity and quasi-affinity problems for Hilbert modules in the Cowen-Douglas class associated with the complex geometric objects, the hermitian anti-holomorphic vector bundles and curvatures. Given a "simple" rank one Cowen-Douglas Hilbert module $\mathcal{M}$, we find necessary and su…

For a ring $R$, we denote by $R[\mathcal L]$ the free $R$-module spanned by the isotopy classes of singular links in $\mathbb S^3$. Given two invertible elements $x,t \in R$, the HOMFLY-PT skein module of singular links in $\mathbb S^3$ (relative to the triple $(R,t,x)$) is the quotient of $R[\mathcal L]$ by local rela…

Let $\{T_1, \ldots, T_n\}$ be a set of $n$ commuting bounded linear operators on a Hilbert space $\mathcal{H}$. Then the $n$-tuple $(T_1, \ldots, T_n)$ turns $\mathcal{H}$ into a module over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \raro \clh, \quad \quad …

Study shows torsion in knot module for specific Montesinos knots.

Researchers compute the Kauffman bracket skein module of a specific 3-manifold.

We make some comments on noncommutative $U(N)$-instantons on $\mathbb{R}^4_θ$. We elaborate on the equations for the ASD-connection for free modules. Further we make some remarks on the computation of the topological index of ADHM instantons.

We compute the equivariant bordism of free oriented $(\mathbb{Z}/p)^n$-manifolds as a module over $Ω_*^{SO}$, when $p$ is an odd prime. We show, among others, that this module is canonically isomorphic to a direct sum of suspensions of multiple tensor products of $Ω^{SO}_*(B \mathbb{Z}/p)$, and that it is generated by …

Develops a new method to study algebraic tangent cones of sheaves using valuations.

The paper defines functions from spherical curves and chord diagrams, proving invariance under specific Reidemeister moves.

Study shows top homology group isn't dualizing module for automorphism group of free groups.

We compute the groups $H^*(\mathrm{Aut}(F_n); M)$ and $H^*(\mathrm{Out}(F_n); M)$ in a stable range, where $M$ is obtained by applying a Schur functor to $H_\mathbb{Q}$ or $H^*_\mathbb{Q}$, respectively the first rational homology and cohomology of $F_n$. For reasons which are not conceptually clear, taking coefficient…

An important part of the classical theory of real or complex manifolds is the theory of (smooth, real analytic or complex analytic) vector bundles. With any vector bundle over a manifold (M,F) the sheaf of its (smooth, real analytic or complex analytic) sections is associated which is a locally free sheaf of F-modules,…

Novel method computes Kauffman bracket skein modules of small 3-manifolds.

Study on cohomology of SL_n(Z) for n>=3, proving vanishing of certain cohomology groups.

We find a presentation of symplectic Steinberg modules and show vanishing cohomology for certain groups.

RB-Modulation trains free diffusion models without external adapters.

Introduces tensor product for quiver representations and applies to stable bundles and character varieties.

We modify our previous construction of link homology in order to include a natural duality functor $\mathfrak{F}$. To a link $L$ we associate a triply-graded module $HXY(L)$ over the graded polynomial ring $R(L)=\mathbb{C}[x_1,y_1,\dots,x_\ell,y_\ell]$. The module has an involution $\mathfrak{F}$ that intertwines the F…

We construct the first combinatorial 1-cocycle with values in the $\mathbb{Z} [x,x^{-1}]$-module of isotopy classes of singular long knots in 3-space with a signed planar double point, and which represents a non trivial cohomology class in the topological moduli space of long knots. It can be interpreted as an invaria…

Study Kauffman bracket skein modules of Seifert fibered spaces.

Khovanov homology distinguishes exotic 4-manifolds.

The spaces of linear differential operators on ${\mathbb{R}}^n$ acting on tensor densities of degree $λ$ and the space of functions on $T^*{\mathbb{R}}^n$ which are polynomial on the fibers are not isomorphic as modules over the Lie algebra $\Vect({\mathbb{R}}^n)$ of vector fields on ${\mathbb{R}}^n$. However, these mo…

We introduce a Lefschetz filtration for integer cohomology and explore its applications.

The objective of this paper is to clarify the relationships between the quantum D-module and equivariant Floer theory. Equivariant Floer theory was introduced by Givental in his paper ``Homological Geometry''. He conjectured that the quantum D-module of a symplectic manifold is isomorphic to the equivariant Floer cohom…

A criterion ensures double sliceness for certain knots and satellite knots.

Study Swan modules and homotopy types, resolving Wall and Dyer questions.

Researchers computed Kauffman bracket skein modules of specific Seifert manifolds.

Study on the dimension of skein modules of Dehn fillings for specific knots.

Study disproves finiteness conjecture for a specific link's skein module.

Study on skein modules and character varieties of Seifert manifolds.

The paper shows geometric realisation over specific groups and knots.

Model proteins with bonds using Kauffman bracket skein module.

Researchers solved a conjecture about a mathematical structure of connected sums of solid tori.

The study proves symplectic quandles cannot have good involutions.

Refined 1-cocycle for knots helps quantify isotopies.

For a prime number $q\neq 2$ and $r>0$ we study, whether there exists an isometry of order $q^r$ acting on a free $\mathbb{Z}_{p^k}$-module equipped with a scalar product. We investigate, whether there exists such an isometry with no non-zero fixed points. Both questions are completely answered in this paper if $p\neq …

We construct an action of the free group $F_n$ on the homotopy category of projective modules over a finite dimensional zigzag algebra. The main theorem in the paper is that this action is faithful. We describe the relationship between homotopy classes of paths in the punctured disc and complexes of projective zigzag m…

We present a complete classification and the construction of $\mathrm{Mp}(2n+2,\mathbb{R})$-equivariant differential operators acting on the principal series representations, associated to the contact projective geometry on $\mathbb{RP}^{2n+1}$ and induced from the irreducible $\mathrm{Mp}(2n,\mathbb{R})$-submodules of…

Study torsion in Kauffman bracket skein modules of 3-manifolds.

The paper proves conditions for self-covering manifolds to be fiber bundles over a circle.

Classifies invariant differential operators on a specific geometric space.

Geometric proof of curve characterization using loop-bundles.

The paper develops residue currents for cohesive modules and proves a generalized Poincaré-Lelong formula.