On the realization of double occurrence words

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).

Title	On the realization of double occurrence words
Creator	Shtylla, B.
	Zulli, Louis P.
	Traldi, Lorenzo
Publisher	Discrete Mathematics
Academic Department	Mathematics
Division	Natural Sciences
Organization	Lafayette College
Date Issued	2009
Date Available	2015-05-06T19:23:05Z
Type	Article
Language	English
Keyword	orthogonal projection
	interlacement graph
	chord diagram
	Gauss code
	double occurrence word
	orthoprojection graph
	circle graph
Bibliographic Citation	Shtylla, B., L. Traldi, and L. P. Zulli (2009) "On the realization of double occurrence words." Discrete Mathematics 309 (6): 1769-73.
Standard Identifier	Handle 10385/1874
Standard Identifier	DOI 10.1016/j.disc.2008.02.035
Permalink	http://hdl.handle.net/10385/1874
Rights Statement	In Copyright
Rights Holders	Elsevier B.V.