arXiv research

It is well known that the opposite F^{op} of the category F of finitely generated free groups is a Lawvere theory for groups, and also that F is a free symmetric monoidal category on a commutative Hopf monoid, or, in other words, a PROP for commutative Hopf algebras. In this paper, we give a direct, combinatorial proof…

New LCM aggregator improves GNN performance and efficiency.

The paper presents new algebraic structures on the 2-sphere using topological field theories.

It is proved that the category of simplicial complete bornological spaces over $\mathbb R$ carries a combinatorial monoidal model structure satisfying the monoid axiom. For any commutative monoid in this category the category of modules is also a monoidal model category with all cofibrant objects being flat. In particu…

In this paper we use 3-manifold techniques to illuminate the structure of the string link monoid. In particular, we give a prime decomposition theorem for string links on two components as well as give necessary conditions for string links to commute under the stacking operation.

The set of homology cobordisms from a surface to itself with markings of their boundaries has a natural monoid structure. To investigate the structure of this monoid, we define and study its Magnus representation and Reidemeister torsion invariants by generalizing Kirk-Livingston-Wang's argument over the Gassner repres…

Under the relation of $0$-concordance, the set of knotted 2-spheres in $S^4$ forms a commutative monoid $\mathcal{M}_0$ with the operation of connected sum. Sunukjian has recently shown that $\mathcal{M}_0$ contains a submonoid isomorphic to $\mathbb{Z}^{\ge 0}$. In this note, we show that $\mathcal{M}_0$ contains a su…

The notion of a quaternionic gerbe is presented as a new way of bundling algebraic structures over a four manifold. The structure groupoid of this fibration is described in some detail. The Euclidean conformal group R*SO(4) appears naturally as a (non-commutative) monoidal structure on this groupoid. Using this monoida…

It is well-known that the monoid of long virtual knots is not commutative. This contrasts with the case of classical long knots, where $A \# B \leftrightharpoons B \# A$ for all $A,B$. In the present paper, we present a new proof that two inequivalent non-classical prime long virtual knots never commute. The original r…

We define the notion of whiskered categories and groupoids, showing that whiskered groupoids have a commutator theory. So also do whiskered $R$-categories, thus answering questions of what might be `commutative versions' of these theories. We relate these ideas to the theory of Leibniz algebras, but the commutator theo…

We use contact geometry to describe the monoid of projectively equivariant meromorphic differential operators on a complex curve, quantization of which generalizes known constructions of classical equivariants to non-commutative function algebras in several variables.

We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on a twin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra (C, W, z, z^*) consist…

The paper develops a new theory of double Johnson filtrations for mapping class groups.

Motivated by the Moore-Segal axioms for an open-closed topological field theory, we consider planar open string topological field theories. We rigorously define a category 2Thick whose objects and morphisms can be thought of as open strings and diffeomorphism classes of planar open string worldsheets. Just as the categ…

Extends manifold theory to $\mathcal{I}$-graded manifolds.

We construct a categorification of the maximal commutative subalgebra of the type $A$ Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The adjoint of this functo…

Study of equivariant Poisson 2-algebra bundles over configuration spaces.

New monoids tied to symmetric group and Jones/Brauer monoids discovered.

In this paper, we introduce a category of graded commutative rings with certain algebraic morphisms, to investigate the cobordism category of plumbed 3-manifolds. In particular, we define a non-associative distributive algebra that gives necessary conditions for an abstract morphism between the homologies of two plumbe…

The singular braids with $n$ strands, $n \geq 3$, were introduced independently by Baez and Birman. It is known that the monoid formed by the singular braids is embedded in a group that is known as singular braid group, denoted by $SG_n$. There has been another generalization of braid groups, denoted by $GVB_n$, $n \ge…

We study monoids generated by Zariski-van Kampen generators in the 17 fundamental groups of the complement of logarithmic free divisors in C^3 listed by Sekiguchi (Theorem 1). Five of them are Artin monoids and eight of them are free abelian monoids. The remaining four monoids are not Gaussian and, hence, are neither G…

In this paper we define a monoid of pseudo braids and prove that this monoid is isomorphic to a singular braid monoid. We also prove an analogue of Markov's theorem for pseudo braids.

Classifies 2D TQFTs for orientable cobordisms using additional data.

The paper introduces a Deligne complex for Artin monoids and studies its properties.

We study algebraic structures of certain submonoids of the monoid of homology cylinders over a surface and the homology cobordism groups, using Reidemeister torsion with non-commutative coefficients. The submonoids consist of ones whose natural inclusion maps from the boundary surfaces induce isomorphisms on higher sol…

We study the algebraic structures of the virtual singular braid monoid, $VSB_n$, and the virtual singular pure braid monoid, $VSP_n$. The monoid $VSB_n$ is the splittable extension of $VSP_n$ by the symmetric group $S_n$. We also construct a representation of $VSB_n$.

This paper gives a self-contained and complete proof of the isomorphism of freely generated monoids extracted from Temperley-Lieb algebras with monoids made of Kauffman's diagrams.

The paper studies properties of Artin monoid Cayley graphs and their quasi-isometry to Deligne complexes.

The paper discusses strictification and non-strictification of monoidal categories.

Smooth actions of the multiplicative monoid $(\mathbb{R},\cdot)$ of real numbers on manifolds lead to an alternative, and for some reasons simpler, definition of a vector bundle, a double vector bundle and related structures like a graded bundle [Grabowski and Rotkiewicz, J. Geom. Phys. 2011]. For these reasons it is n…

The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in the simplicial category, are submonoids of monoids one finds in Temperley-Lieb algebras, and as the monoids of Temperley-Lieb algebras are linked to situations where an endofunctor is adjoint to itself, so the monoids of simplicial endomorphi…

The paper constructs semistrict monoidal 2-categories from foam evaluations.

Study groups formed by words in braid monoid.

The main new notions are the notions of tangent-like spaces and local monoids. The main result is the pasage from a local monoid to its tangent-like space which is a local Leibniz algebra.

In this paper we show that the singular braid monoid of an orientable surface can be embedded in a group. The proof is purely topological, making no use of the monoid presentation.

Extends Jones' construction to map Thompson group to pointed links.

We show that a certain linear representation of the singular braid monoid on three strands is faithful. Furthermore we will give a second - group theoretically motivated - solution to the word problem in this monoid.

3D HQFTs constructed using graded monoidal categories.

We give a presentation of the $n$-dimensional oriented cobordism category $\text{Cob}_n$ with generators corresponding to diffeomorphisms and surgeries along framed spheres, and a complete set of relations. Hence, given a functor $F$ from the category of smooth oriented manifolds and diffeomorphisms to an arbitrary cat…

The paper studies submonoids of singular twisted virtual braids and their properties.

Monoidal categorifies genus zero skein algebra using K-theory.

We initiate the study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary. The monoid strictly contains the monoid of products of positive Dehn twists. We explain the relationship to tight contact structures and open book decompositions.

Abstract: Generalizes Milnor-Schwarz lemma to inverse monoids.

Garside's results and the existense of the greedy normal form for braids are shown to be true for the singular braid monoid. An analogue of the presentation of J. S. Birman, K. H. Ko and S. J. Lee for the braid group is also obtained for this monoid.

Study on endomorphism and automorphism groups of specific quandles.

We give presentations, in terms of generators and relations, for the monoids of singular braids on closed surfaces. The proof of the validity of these presentations can also be applied to verify, in a new way, the presentations given by Birman for the monoids of Singular Artin braids.

In this paper we study the growth rates of Artin monoids and we show that 4 is a universal upper bound. We also show that the generating functions of the associated right-angled Artin monoids are given by families of Chebyshev polynomials. Applications to Artin groups and positive braids are given.

A homology cylinder over a surface consists of a homology cobordism between two copies of the surface and markings of its boundary. The set of isomorphism classes of homology cylinders over a fixed surface has a natural monoid structure and it is known that this monoid can be seen as an enlargement of the mapping class…