[Paper Review] Experimental Mathematics Approach to Gauss Diagrams Realizability

Abdullah Khan, Alexei Lisitsa+1 more|arXiv (Cornell University)|Mar 3, 2021

Geometric and Algebraic Topology21 references4 citations

TL;DR

This paper uses experimental mathematics and Prolog-based constraint satisfaction to investigate realizability criteria for Gauss diagrams—combinatorial encodings of planar curves. The authors implement and test recent declarative criteria from [Bir19] and [GL18, GL20], discovering that these criteria are incorrect: they identify non-realizable diagrams as realizable, with a minimal counterexample at size 9 and 6 and 36 counterexamples at sizes 10 and 11, respectively.

ABSTRACT

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.

Motivation & Objective

• To investigate the correctness of recently proposed declarative realizability criteria for Gauss diagrams using computational experimentation.
• To enumerate non-equivalent Gauss diagrams up to size 11 using logic programming and canonical lintel representation.
• To test the equivalence of various realizability conditions, including classical Dehn’s criterion and modern parity-based conditions.
• To identify and validate counterexamples where modern criteria fail to detect non-realizable diagrams.
• To assess the logical definability of realizability in first-order logic with parity quantifiers over interlacement graphs.

Proposed method

• Represent Gauss diagrams as 'lintels'—canonical permutations of chord endpoints—using a bijection β to generate all non-equivalent diagrams.
• Implement Prolog-based constraint satisfaction to generate and canonize lintels efficiently using built-in permutation and dynamic fact storage.
• Integrate and test multiple realizability criteria: classical Dehn (CA), STZ, B (Bir19), and GL (GL18, GL20) conditions.
• Use the interlacement graph of each diagram to evaluate parity-based conditions in FO+MOD₂ logic, enabling efficient polynomial-time checks.
• Apply backtracking and dynamic fact storage to avoid duplicates and ensure complete enumeration of non-equivalent diagrams.
• Compare results across criteria and validate against the classical Dehn algorithm to detect discrepancies.

Experimental results

Research questions

• RQ1 Do the recently proposed realizability criteria in [Bir19] and [GL18, GL20] correctly identify all realizable Gauss diagrams?
• RQ2 Are the B and GL criteria logically equivalent to the classical Dehn criterion and the STZ condition?
• RQ3 Can counterexamples be systematically generated and enumerated using logic programming and lintel representation?
• RQ4 At what diagram size do discrepancies first appear between the B/GL criteria and the true realizability condition?
• RQ5 Is realizability of Gauss diagrams definable in first-order logic with parity quantifiers over interlacement graphs?

Key findings

• The B and GL realizability criteria from [Bir19] and [GL18, GL20] are incorrect: they classify non-realizable diagrams as realizable.
• A minimal counterexample exists at size 9, with exactly one non-equivalent diagram satisfying B and GL conditions but failing realizability.
• At size 10, six non-equivalent diagrams satisfy B and GL conditions but are not realizable, increasing to 36 at size 11.
• The classical Dehn criterion (CA) and STZ condition remain equivalent and correct up to size 11, matching the true count of realizable diagrams.
• The B and GL criteria diverge from CA and STZ starting at size 9, indicating they are not sound realizability conditions.
• The authors conjecture that Gauss diagram realizability cannot be fully captured in first-order logic with parity quantifiers over interlacement graphs.