Related papers: On the Categorified Wrapping Number Conjecture

The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space (defined in [Trunks and classifying spaces, Applied Categorical Structures, 3 (1995)…

We survey some recent results concerning the so called Categorical Torelli problem. This is to say how one can reconstruct a smooth projective variety up to isomorphism, by using the homological properties of special admissible…

We prove the "Gluing Conjecture" on the spectral side of the categorical geometric Langlands correspondence. The key tool is the structure of crystal on the category of singularities, which allows to reduce the conjecture to the question of…

A braided $Ann$-category $\mathcal A$ is an $Ann$-category $\mathcal A$ together with a braiding $c$ such that $(\mathcal A, \otimes, a, c, (1,l,r))$ is a braided tensor category, moreover $c$ is compatible with the distributivity…

We prove that Legendrian and transverse links in overtwisted contact structures having overtwisted complements can be classified coarsely by their classical invariants. We further prove that any coarse equivalence class of loose links has…

This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof…

The notion of Ann-categories is a categorification of the ring structure. Regular Ann-categories were classified by Shukla algebraic cohomology. In this article, we state and prove the precise theorem on classification for the general case…

This paper gives new and elementary combinatorial topological proofs of the classification of unoriented and oriented rational knots and links. These proofs are based on the known classification of alternating knots through flyping, and the…

For any Legendrian link in $\displaystyle \mathbb{R}^{3}$ given by the rainbow closure of a positive braid word, we develop an explicit and computable description of a Legendrian isotopy invariant associated with it, namely the…

We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative…

This paper has partially a novel and partially a survey character. We start with a short review of rack (two term) homology of self distributive algebraic structures (shelves) and their connections to knot theory. We concentrate on a…

King's conjecture states that on every smooth complete toric variety $X$ there exists a strongly exceptional collection which generates the bounded derived category of $X$ and which consists of line bundles. We give a counterexample to this…

A classification of spanning surfaces for alternating links is provided up to genus, orientability, and a new invariant that we call aggregate slope. That is, given an alternating link, we determine all possible combinations of genus,…

In this paper we present a categorical version of the first and second fundamental theorems of the invariant theory for the quantized symplectic groups. Our methods depend on the theory of braided strict monoidal categories which are…

Let $\mathcal{C}$ be an additive category equipped with an automorphism $\Sigma$. We show how to obtain $n$-angulations of $(\mathcal{C},\Sigma)$ using some particular periodic injective resolutions. We give necessary and sufficient…

Let $R$ be a commutative ring If $\mathcal{C}_1$ and $\mathcal{C}_2$ are $R$-linear triangulated categories then we can give an obvious triangulated structure on $\mathcal{C} = \mathcal{C}_1 \oplus \mathcal{C}_2$ where $Hom_\mathcal{C}(U,…

We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context.…

We develop a calculus for diagrams of knotted objects. We define Arrow presentations, which encode the crossing informations of a diagram into arrows in a way somewhat similar to Gauss diagrams, and more generally w-tree presentations,…

Using methods from coarse topology we show that fundamental classes of closed enlargeable manifolds map non-trivially both to the rational homology of their fundamental groups and to the K-theory of the corresponding reduced C*-algebras.…

The third author has shown that Shelah's eventual categoricity conjecture holds in universal classes: class of structures closed under isomorphisms, substructures, and unions of chains. We extend this result to the framework of…