On the realization of double occurrence words

TL;DR

This paper explores the conditions under which double occurrence words can be realized by closed curves in the plane, providing new graph-theoretic and algebraic characterizations, including an elegant algebraic criterion involving idempotent matrices over GF(2).

Contribution

It introduces a novel algebraic characterization of realizable double occurrence words using idempotent matrices over GF(2), extending previous graph-theoretic results.

Findings

A double occurrence word is realizable iff there exists a diagonal matrix D_S making M_S+D_S idempotent over GF(2).

Provides new graph-theoretic criteria for realizability.

Reformulates classical results with algebraic and matrix-theoretic approaches.

Abstract

Let S be a double occurrence word, and let M_S be the word's interlacement matrix, regarded as a matrix over GF(2). Gauss addressed the question of which double occurrence words are realizable by generic closed curves in the plane. We reformulate answers given by Rosenstiehl and by de Fraysseix and Ossona de Mendez to give new graph-theoretic and algebraic characterizations of realizable words. Our algebraic characterization is especially pleasing: S is realizable if and only if there exists a diagonal matrix D_S such that M_S+D_S is idempotent over GF(2).