Representation of finite connective spaces

TL;DR

This paper explores representing finite connectivity spaces using directed graphs and links, showing certain classes can be represented and conjecturing all finite spaces can be represented by links.

Contribution

It introduces a method to represent finite connectivity spaces with directed graphs and links, and demonstrates this for specific classes, proposing a universal conjecture.

Findings

Finite connectivity spaces can be represented by directed simple graphs.

All iterated Brunnian spaces admit link representations.

Conjecture: every finite connectivity space can be represented by a link.

Abstract

After recalling the definition of connectivity spaces and some of their main properties, a way is proposed to represent finite connectivity spaces by directed simple graphs. Then a connectivity structure is associated to each tame link. It is showed that all spaces of a certain class (the iterated Brunnian ones) admit representations by links. Finally, I conjecture that every finite connectivity space is representable by a link. ----- Apres un rappel de la definition des espaces connectifs et de certaines de leurs principales proprietes, nous proposons une maniere de representer les espaces connectifs finis par des graphes simples orientes, puis nous associons a tout entrelacs une structure connective. Nous montrons que tout espace d'une certaine classe (les espaces brunniens iteres) admet une representation par entrelacs, et nous conjecturons finalement que tout espace connectif fini est representable par entrelacs.