[Paper Review] Circle graphs (chord interlacement graphs) of Gauss diagrams: Descriptions of realizable Gauss diagrams, algorithms, enumeration

Abdullah Khan, Alexei Lisitsa+2 more|arXiv (Cornell University)|Aug 5, 2021

Geometric and Algebraic Topology24 references4 citations

TL;DR

This paper re-examines realizability criteria for Gauss diagrams using circle graphs (chord interlacement graphs), identifies flaws in prior theoretical descriptions through experimental validation, proposes corrected realizability criteria, and introduces an efficient algorithm for checking realizability. It enumerates realizable Gauss diagrams and meander graphs up to size 13, linking the counts to classes of mutant alternating knots and contributing new sequences to the OEIS, including A343358 for Gauss graphs and A338660 for meander graphs.

ABSTRACT

Chord diagrams, under the name of Gauss diagrams, are used in low-dimensional topology as an important tool for studying curves or knots. Those Gauss diagrams that correspond to curves or knots are called realizable. The theme of our paper is the fact that realizability of a Gauss diagram can be expressed via its circle graph. Accordingly, one can define and study realizable circle graphs (with realizability of a circle graph understood as realizability of any one of chord diagrams corresponding to the graph). Several studies contain theorems purporting to prove the fact. We check several of these descriptions experimentally and find counterexamples to the descriptions of realizable Gauss diagrams in some of these publications. We formulate new descriptions of realizable circle graphs and present an elegant algorithm for checking if a circle graph is realizable. We enumerate realizable circle graphs for small sizes and comment on these numbers. Then we concentrate on one type of curves, called meanders, and study the circle graphs of their Gauss diagrams.

Motivation & Objective

• To investigate the consistency between realizability of Gauss diagrams and their corresponding circle graphs.
• To identify and correct errors in previously published realizability criteria for Gauss diagrams.
• To develop an efficient algorithm for determining whether a circle graph is realizable.
• To enumerate realizable Gauss diagrams and meander diagrams for small sizes and analyze the resulting sequences.
• To explore the relationship between circle graphs of realizable Gauss diagrams and classes of mutant alternating knots.

Proposed method

• The authors use experimental validation to test existing realizability criteria for Gauss diagrams by generating and analyzing all non-isomorphic Gauss diagrams up to size 13.
• They implement a permutation-based algorithm and an incremental algorithm to generate all non-equivalent Gauss diagrams, up to size 11 and 12 respectively, and use Tait Curves for size 13.
• Circle graphs are constructed from Gauss diagrams by treating chords as vertices and connecting them if they intersect in the diagram.
• An isomorphism check using the NetworkX library in Python is applied to classify and count non-isomorphic Gauss graphs and meander graphs.
• The method cross-validates results with known knot theory data, particularly Stoimenov’s knot tables, to verify counts against mutant knot equivalence classes.
• Theoretical results are used to link circle graph isomorphism to knot mutation moves, establishing that isomorphic circle graphs correspond to diagrams related by mutation.

Experimental results

Research questions

• RQ1 Are the realizability criteria for Gauss diagrams described in prior literature accurate and complete?
• RQ2 Can an efficient algorithm be developed to determine whether a given circle graph is realizable?
• RQ3 What are the exact counts of realizable Gauss diagrams and meander diagrams for small sizes, and how do they relate to knot invariants?
• RQ4 Is there a direct correspondence between the number of non-isomorphic circle graphs of realizable Gauss diagrams and the number of mutant knot classes?
• RQ5 Do the sequences of realizable Gauss graphs and meander graphs match known sequences in knot theory, such as alternating knots modulo mutation?

Key findings

• The paper identifies counterexamples to previously published realizability criteria, demonstrating that some theoretical descriptions are incorrect.
• The authors propose new, corrected realizability criteria for circle graphs based on experimental validation and theoretical consistency.
• An efficient algorithm for checking realizability of a circle graph is developed and implemented, enabling enumeration up to size 13.
• The number of non-isomorphic realizable Gauss graphs (A343358) matches the number of alternating prime knots up to size 10, but diverges at size 11 due to mutant knot pairs.
• The sequence A343358 is confirmed to count both the number of non-isomorphic circle graphs of realizable Gauss diagrams and the number of mutant knot classes, validating the theoretical link to knot mutation.
• The count of realizable meander graphs (A338660) is shown to be consistent with known data, and the method contributes new terms to OEIS, enabling cross-verification with knot theory databases.