Journal of Combinatorics

Volume 15 (2024)

Number 1

Duality and tangles of set separations

Pages: 1 – 39

DOI: https://dx.doi.org/10.4310/JOC.2024.v15.n1.a1

Authors

Diestel Reinhard (Department of Mathematics, Universität Hamburg, Germany)

Christian Elbracht (Department of Mathematics, Universität Hamburg, Germany)

Joshua Erde (Institute of Discrete Mathematics, Graz University of Technology, Graz, Austria)

Maximilian Teegen (Department of Mathematics, Universität Hamburg, Germany)

Abstract

Applications of tangles of connectivity systems suggest a duality between these, in which for two sets $X$ and $Y$ the elements $x$ of $X$ map to subsets $Y_x$ of $Y$, and the elements $y$ of $Y$ map to subsets $X_y$ of $X$, so that $x \in X_y$ if and only if $y \in Y_x$ for all $x \in X$ and $y \in Y$. We explore this duality, and relate the tangles arising from the dual systems to each other.

Keywords

tangles, homology

2010 Mathematics Subject Classification

05C83

Received 20 September 2021

Accepted 7 July 2022

Published 7 November 2023