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Experimental Mathematics Approach to Gauss Diagrams Realizability

A Gauss diagram (or, more generally, a chord diagram) consists of a circle and some chords inside it. Gauss diagrams are a well-established tool in the study of topology of knots and of planar and spherical curves. Not every Gauss diagram corresponds to a knot (or an immersed curve); if it does, it is called realizable. A classical question of computational topology asked by Gauss himself is which chords diagrams are realizable. An answer was first discovered in the 1930s by Dehn, and since then many efficient algorithms for checking realizability of Gauss diagrams have been developed. Recent studies in Grinblat-Lopatkin (2018,2020) and Biryukov (2019) formulated especially simple conditions related to realizability which are expressible in terms of parity of chords intersections. The simple form of these conditions opens an opportunity for experimental investigation of Gauss diagrams using constraint satisfaction and related techniques. In this paper we report on our experiments with Gauss diagrams of small sizes (up to 11 chords) using implementations of these conditions and other algorithms in logic programming language Prolog. In particular, we found a series of counterexamples showing that that realizability criteria established by Grinblat and Lopatkin (2018,2020) and Biryukov (2019) are not completely correct.