Related papers: The augmentation category map induced by exact Lag…

A folklore result in category theory is that a (weakly) Cartesian closed category with finite co-products is distributive. Usually, the proof of this small result is carried on using the fact that the exponential functor is right adjoint to…

The augmentation variety of a knot is the locus, in the 3-dimensional coefficient space of the knot contact homology dg-algebra, where the algebra admits a unital chain map to the complex numbers. We explain how to express the Alexander…

In this paper, we first provide an explicit procedure to glue together hereditary exact model structures for the recollement of exact categories. To that end, we use the notion of cotorsion pairs and we investigate the gluing of complete…

The problem of classification of Legendrian knots (links) up to isotopy in the class of Legendrian embeddings (Legendrian isotopy) naturally leads to the following two subproblems. The first of them is: which combinations of the three…

It is proved in \cite{BP} (arXiv:1108.2717) that the category of relative 3-dimensional cobordisms $\cal Cob^{2+1}$ is equivalent to the universal algebraic category $\overline{\overline{\cal H}}{}^r$ generated by a Hopf algebra object. A…

We show that a two-dimensional totally real concordance can be approximated by a Lagrangian concordance whose Legendrian boundary has been stabilised both positively and negatively sufficiently many times. The main applications that we…

We show that direct summands of certain additive functors arising as bifunctors with a fixed argument in an abelian category are again of that form whenever the fixed argument has finite length or, more generally, satisfies the descending…

Let $R$ be an integral domain and $G$ be a subgroup of its group of units. We consider the category $\mathbf{\mathsf{Cob}}_G$ of 3-dimensional cobordisms between oriented surfaces with connected boundary, equipped with a representation of…

We provide an explicit example of a non trivial Legendrian knot $\Lambda$ such that there exists a Lagrangian concordance from $\Lambda_0$ to $\Lambda$ where $\Lambda_0$ is the trivial Legendrian knot. We then use the map induced in…

This work studies Legendrian loop actions on exact Lagrangian fillings of Legendrian links in $(\R^3, \xi_{\st})$. By identifying the induced action of Legendrian loops as generators of cluster modular groups, we establish the existence of…

For a stably framed Liouville manifold X , we construct a "Donaldson-Fukaya category over the sphere spectrum" F(X; S). The objects are closed exact Lagrangians whose Gauss maps are nullhomotopic compatibly with the ambient stable framing,…

We consider knot invariants in the context of large $N$ transitions of topological strings. In particular we consider aspects of Lagrangian cycles associated to knots in the conifold geometry. We show how these can be explicity constructed…

In this note, we first classify all topological torus knots lying on the Heegaard torus in lens spaces, and then we study Legendrian representatives of these knots. We classify oriented positive Legendrian torus knots in the universally…

The cartesian structure possessed by relations, spans, profunctors, and other such morphisms is elegantly expressed by universal properties in double categories. Though cartesian double categories were inspired in part by the older program…

In this article we extend evaluations of the Kauffman bracket on regular isotopy classes of knots and links to a variety of functors defined on the category of framed tangles. We show that many such functors exist, and that they correspond…

We present an efficient and user-friendly method for constructing any cofibrantly generated model structure on the category of double categories whose trivial fibrations are the "canonical" ones: the double functors which are surjective on…

We demonstrate that any full and faithful $*$-functor between approximable categories of locally finite coarse spaces induces a coarse embedding between the underlying spaces. Furthermore, we establish a general characterisation of such…

Given a double category D such that D_0 has pushouts, we characterize oplax/lax adjunctions between D and Cospan(D_0) such that the right adjoint is normal and restricts to the identity on D_0, where Cospan(D_0) denotes the double category…

We characterize which Legendrian $4$-plat knots in the standard contact $3$-space have exact orientable Lagrangian fillings. As a corollary, we show that the underlying smooth knot types of fillable Legendrian $4$-plats are positive.

We provide bar and cobar constructions as functors between some categories of curved algebras and curved augmented coalgebras over a graded commutative ring. These functors are adjoint to each other.