On the realization of double occurrence words |
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Authors: | B. Shtylla |
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Affiliation: | a Department of Mathematics, University of Utah, Salt Lake City, UT 84112, United States b Department of Mathematics, Lafayette College, Easton, PA 18042, United States |
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Abstract: | Let S be a double occurrence word, and let MS be the word’s interlacement matrix, regarded as a matrix over . 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 DS such that MS+DS is idempotent over . |
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Keywords: | Double occurrence word Gauss code Interlacement graph Orthoprojection graph Orthogonal projection Circle graph Chord diagram |
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