Handle slides for delta-matroids

TL;DR

This paper extends the classical surface handle-slide classification to binary delta-matroids, establishing a canonical form analogous to surface decompositions using delta-matroid handle-slides.

Contribution

It introduces a delta-matroid analogue of handle-slides and provides a canonical form for binary delta-matroids similar to surface classifications.

Findings

Binary delta-matroids can be decomposed into a direct sum of simpler delta-matroids.

The delta-matroid classification aligns with surface topology results.

The work bridges ribbon graph theory and delta-matroid theory.

Abstract

A classic exercise in the topology of surfaces is to show that, using handle slides, every disc-band surface, or 1-vertex ribbon graph, can be put in a canonical form consisting of the connected sum of orientable loops, and either non-orientable loops or pairs of interlaced orientable loops. Motivated by the principle that ribbon graph theory informs delta-matroid theory, we find the delta-matroid analogue of this surface classification. We show that, using a delta-matroid analogue of handle-slides, every binary delta-matroid in which the empty set is feasible can be written in a canonical form consisting of the direct sum of the delta-matroids of orientable loops, and either non-orientable loops or pairs of interlaced orientable loops. Our delta-matroid results are compatible with the surface results in the sense that they are their ribbon graphic delta-matroidal analogues.