Caracteritzant els codis de Gauss: només cal girar la cantonada

Abstract

In this paper we give some historical background, the motivation and some solutions to the Gauss codes problem. A closed curve on the plane with n self-intersection points that are nontangential, and where the curve goes through all points of the plane twice at the most, generates a double occurrence word on n symbols as follows. Label the intersection points with f1; : : : ;ng, choose an initial point on the curve, a direction of travel through the curve, and record the sequence of selfintersection points found. Such word is said to be a Gauss code representable on the plane. The Gauss codes problem seeks to characterize those double occurrence words that are representable by plane curves. We explain two different characterizations that involve turning the corner' to undo the crossing points in the two most natural ways. The first one leads to the characterization known on account of Dehn, and Read and Rosenstiehl. We present an alternative characterization that uses the other way of undoing' the crossing points and involves the Seifert cycles of the word. We also give some historical insight.