Related papers: Vassiliev theory and regional change

Essentials of sheaves are briefly presented, followed by related comments on presheaves, bundles, manifolds and singularities, aiming to point to their differences not only in their different formal mathematical structures, but also in the…

Consider a given space, e.g., the Euclidean plane, and its decomposition into Voronoi regions induced by given sites. It seems intuitively clear that each point in the space belongs to at least one of the regions, i.e., no neutral region…

Convolution is an important tool in the construction of positive definite kernels on a manifold. This contribution provides conditions on an $L^2$-positive definite and zonal kernel on the unit sphere of $\mathbb{C}^q$ in order that the…

These notes are meant to provide a rapid introduction to triangulated categories. We start with the definition of an additive category and end with a glimps of tilting theory. Some exercises are included.

We exhibit an encoding of knots into processes in the {\pi}-calculus such that knots are ambient isotopic if and only their encodings are weakly bisimilar.

We define a knot invariant and a 2-knot invariant from any finite categorical group. We calculate an explicit example for the Spun Trefoil.

Many classic questions of structural theory concern discrete changes, such as the formation or dissolution of groups, role turnover, or faction realignment. Here, we consider a basic framework combining prior work on change paths and recent…

Invariant theory is concerned with functions that do not change under the action of a given group. Here we communicate an approach based on tensor networks to represent polynomial local unitary invariants of quantum states. This graphical…

The aim of this paper is to define certain algebraic structures coming from generalized Reidemeister moves of singular knot theory. We give examples, show that the set of colorings by these algebraic structures is an invariant of singular…

A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded…

This article offers an intuitive introduction to monoidal categories through the lens of painting, presenting abstract mathematical concepts with visual and tactile analogies. Aimed at curious undergraduates and non-specialists, it seeks to…

The objective of this work is twofold. On one hand, it is intended as a short introduction to spin networks and invariants of 3-manifolds. It covers the main areas needed to have a first understanding of the topics involved in the…

Weight systems are functions on chord diagrams satisfying Vassiliev's $4$-term relations. They originate in the theory of finite type knot invariants. Recent developments in understanding weight systems arising from Lie algebras are based…

This paper considers systems subject to nonholonomic constraints which are not uniform on the whole configuration manifold. When the constraints change, the system undergoes a transition in order to comply with the new imposed conditions.…

Parity mappings from the chords of a Gauss diagram to the integers is defined. The parity of the chords is used to construct families of invariants of Gauss diagrams and virtual knots. One family consists of degree $n$ Vassiliev invariants.

The structure of covariant instruments is studied and a general structure theorem is derived. A detailed characterization is given to covariant instruments in the case of an irreducible representation of a locally compact group.

Adopting a purely group-theoretical point of view, we consider the star product of functions which is associated, in a natural way, with a square integrable (in general, projective) representation of a locally compact group. Next, we show…

The Kakimizu complex is usually defined in the context of knots, where it is known to be quasi-Euclidean. We here generalize the definition of the Kakimizu complex to surfaces and 3-manifolds (with or without boundary). Interestingly, in…

Sometimes we obtain attractive results when associating facts to simple elements. The goal of this work is to introduce a possible alternative in the study of the dynamics of rational maps.

We explain how to define the Robinson-Schensted-Knuth (RSK) correspondence in terms of local transformations called "toggles." (This note, which is not intended for publication and which is based on presentations of Alex Postnikov, was…