A note on geometric duality in matroid theory and knot theory

TL;DR

This paper explores the connection between geometric duality in planar graphs and its implications for knot theory, revealing that certain graph isomorphisms correspond to link equivalences.

Contribution

It establishes a novel link between duality in planar graphs and equivalence relations in knot theory, highlighting a surprising correspondence.

Findings

Geometric duality generates both 2-isomorphism and abstract duality in planar graphs.

Checkerboard graph isomorphisms correspond to 2-isomorphisms in links.

The equivalence relations in knot theory can be characterized via graph dualities.

Abstract

We observe that for planar graphs, the geometric duality relation generates both 2-isomorphism and abstract duality. This observation has the surprising consequence that for links, the equivalence relation defined by isomorphisms of checkerboard graphs is the same as the equivalence relation defined by 2-isomorphisms of checkerboard graphs.