On Connectivity Spaces

TL;DR

This paper explores the fundamental properties of connectivity spaces, including their categorical structures, homotopy, and invariants, and introduces a new link invariant called the connectivity order.

Contribution

It provides new insights into the categorical and algebraic structures of connectivity spaces and introduces the connectivity order as a novel link invariant.

Findings

Connectivity spaces form closed monoidal categories with tensor and smash products.

Finite connectivity spaces can be represented by links via the Brunn-Debrunner-Kanenobu theorem.

A new numerical invariant, the connectivity order, is defined for links.

Abstract

This paper presents some basic facts about the so-called connectivity spaces. In particular, it studies the generation of connectivity structures, the existence of limits and colimits in the main categories of connectivity spaces, the closed monoidal category structure given by the so-called tensor product on integral connectivity spaces; it defines homotopy for connectivity spaces and mention briefly related difficulties; it defines smash product of pointed integral connectivity spaces and shows that this operation results in a closed monoidal category with such spaces as objects. Then, it studies finite connectivity spaces, associating a directed acyclic graph with each such space and then defining a new numerical invariant for links: the connectivity order. Finally, it mentions the not very wellknown Brunn-Debrunner-Kanenobu theorem which asserts that every finite integral connectivity space can be represented by a link.