On the realization of double occurrence words

Let SS be a double occurrence word, and let MSMS 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: SS is realizable if and only if there exists a diagonal matrix DSDS such that MS+DSMS+DS is idempotent over GF(2).