On realizabilty of Gauss diagrams and constructions of meanders

Grinblat, Andrey; Lopatkin, Viktor

doi:10.1142/s0218216520500315

Cited by 4 publications

(11 citation statements)

References 10 publications

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“…We have found the minimal counterexample of size n=9 for both these descriptions (first reported in [KLV21a]), and also enumerated counterexamples for n=10 and 11. We reflect on the counterexamples and highlight the error in the arguments of [GL18,GL20] and [Bir19]. We provide a correction and propose new complete realizability criteria.…”

Section: Theorem 1 a Graph Is Realizable If And Only If It Correspond...mentioning

confidence: 96%

“…B and GL, however are not equivalent to CA or STZ, starting from size 9. Hence B and GL are not correct descriptions of realizability, despite the claims in [Bir19] and [GL18,GL20], respectively.…”

Section: Diagrams Generation Experiments and Counterexamplesmentioning

confidence: 96%

“…Another desciption for realizability expressible in terms of interlacement graph was proposed by Grinblat and Lopatkin in [GL18,GL20]. It is claimed that a prime Gauss diagram g is realizable if and only if the following conditions hold true:…”

Section: Gl-conditionsmentioning

confidence: 99%

“…Also in [GL20] it was announced that if a Gauss diagram G is realizable (say by a plane curve C (G)), then to every closed path (say) P along C (G) we can associate a colouring of another part of C (G) into two colours (roughly speaking we get "inner" and "outer" sides of P, cf. Jordan curve Theorem).…”

Section: What Was Wrong In the Gl Description Of Realizabilitymentioning

confidence: 99%

“…Then in [GL20] it was shown that in the case of non realizability of a Gauss diagram we always can find a special kind of such counters (X-counter), in terms of Gauss diagrams this counter is made by two intersecting chords. The counterexamples in Section 4 show that this statement fails.…”

Section: What Was Wrong In the Gl Description Of Realizabilitymentioning

confidence: 99%

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Circle graphs (chord interlacement graphs) of Gauss diagrams: Descriptions of realizable Gauss diagrams, algorithms, enumeration

Khan,

Lisitsa,

Lopatkin

et al. 2021

Preprint

Self Cite

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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.

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Section: Theorem 1 a Graph Is Realizable If And Only If It Correspond...mentioning

confidence: 96%

Section: Diagrams Generation Experiments and Counterexamplesmentioning

confidence: 96%

Section: Gl-conditionsmentioning

confidence: 99%

Section: What Was Wrong In the Gl Description Of Realizabilitymentioning

confidence: 99%

Section: What Was Wrong In the Gl Description Of Realizabilitymentioning

confidence: 99%

See 3 more Smart Citations

Circle graphs (chord interlacement graphs) of Gauss diagrams: Descriptions of realizable Gauss diagrams, algorithms, enumeration

Khan,

Lisitsa,

Lopatkin

et al. 2021

Preprint

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Gauss-Lintel, an Algorithm Suite for Exploring Chord Diagrams

Khan

Lisitsa

Vernitski

2021

Lecture Notes in Computer Science

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Describing realizable Gauss diagrams using the concepts of parity or bipartite graphs

Lisitsa¹,

Lopatkin²,

Vernitski³

2023

J. Knot Theory Ramifications

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Two recent publications describe realizable Gauss diagrams using conditions stating that the number of chords in certain sets of chords is even or odd. We demonstrate that these descriptions are incorrect by finding multiple counter-examples. However, the idea of having a parity-based description of realizable Gauss diagrams is attractive. We recall that realizability of Gauss diagrams as touch curves can be described via bipartite graphs. We show that realizable Gauss diagrams can be described via bipartite graphs.

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On realizabilty of Gauss diagrams and constructions of meanders

Cited by 4 publications

References 10 publications

Circle graphs (chord interlacement graphs) of Gauss diagrams: Descriptions of realizable Gauss diagrams, algorithms, enumeration

Circle graphs (chord interlacement graphs) of Gauss diagrams: Descriptions of realizable Gauss diagrams, algorithms, enumeration

Gauss-Lintel, an Algorithm Suite for Exploring Chord Diagrams

Describing realizable Gauss diagrams using the concepts of parity or bipartite graphs

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