Counting graphs induced by Gauss diagrams and families of mutant alternating knots
The construction known as Gauss diagrams or Gauss words is one of the oldest in knot theory and has been studied extensively both in the context of knots and in the context of closed curves with self-intersections. When we studied graphs induced by Gauss diagrams, we produced all examples of these g...
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| Format: | Article |
| Language: | English |
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Elsevier
2024-12-01
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| Series: | Examples and Counterexamples |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2666657X24000284 |
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| author | Alexei Lisitsa Alexei Vernitski |
| author_facet | Alexei Lisitsa Alexei Vernitski |
| author_sort | Alexei Lisitsa |
| collection | DOAJ |
| description | The construction known as Gauss diagrams or Gauss words is one of the oldest in knot theory and has been studied extensively both in the context of knots and in the context of closed curves with self-intersections. When we studied graphs induced by Gauss diagrams, we produced all examples of these graphs of small sizes, and we published the number of these graphs as sequence A343358 in the OEIS. The aim of this article is to clarify several subtle theoretical points concerning A343358. Most importantly, we explain why our numbers, produced using graph-theoretical constructions, reflect the number of so-called mutant knots. |
| format | Article |
| id | doaj-art-9e64ce6263fa4db7aef2337b2a435b51 |
| institution | OA Journals |
| issn | 2666-657X |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Examples and Counterexamples |
| spelling | doaj-art-9e64ce6263fa4db7aef2337b2a435b512025-08-20T01:58:31ZengElsevierExamples and Counterexamples2666-657X2024-12-01610016210.1016/j.exco.2024.100162Counting graphs induced by Gauss diagrams and families of mutant alternating knotsAlexei Lisitsa0Alexei Vernitski1Department of Computer Science, University of Liverpool, Ashton building, Liverpool, L69 3DR, UKSchool of Mathematics, Statistics and Actuarial Science, University of Essex, Colchester, CO4 3SQ, UK; Corresponding author.The construction known as Gauss diagrams or Gauss words is one of the oldest in knot theory and has been studied extensively both in the context of knots and in the context of closed curves with self-intersections. When we studied graphs induced by Gauss diagrams, we produced all examples of these graphs of small sizes, and we published the number of these graphs as sequence A343358 in the OEIS. The aim of this article is to clarify several subtle theoretical points concerning A343358. Most importantly, we explain why our numbers, produced using graph-theoretical constructions, reflect the number of so-called mutant knots.http://www.sciencedirect.com/science/article/pii/S2666657X24000284Gauss diagramInterlacement graphCircle graphAlternating knotMutant knot |
| spellingShingle | Alexei Lisitsa Alexei Vernitski Counting graphs induced by Gauss diagrams and families of mutant alternating knots Examples and Counterexamples Gauss diagram Interlacement graph Circle graph Alternating knot Mutant knot |
| title | Counting graphs induced by Gauss diagrams and families of mutant alternating knots |
| title_full | Counting graphs induced by Gauss diagrams and families of mutant alternating knots |
| title_fullStr | Counting graphs induced by Gauss diagrams and families of mutant alternating knots |
| title_full_unstemmed | Counting graphs induced by Gauss diagrams and families of mutant alternating knots |
| title_short | Counting graphs induced by Gauss diagrams and families of mutant alternating knots |
| title_sort | counting graphs induced by gauss diagrams and families of mutant alternating knots |
| topic | Gauss diagram Interlacement graph Circle graph Alternating knot Mutant knot |
| url | http://www.sciencedirect.com/science/article/pii/S2666657X24000284 |
| work_keys_str_mv | AT alexeilisitsa countinggraphsinducedbygaussdiagramsandfamiliesofmutantalternatingknots AT alexeivernitski countinggraphsinducedbygaussdiagramsandfamiliesofmutantalternatingknots |