A Study of Gram Determinants in Knot Theory
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909927178502144 |
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| author | Ibarra, Dionne Montoya-Vega, Gabriel |
| author_facet | Ibarra, Dionne Montoya-Vega, Gabriel |
| contents | Historically originated as a sub-field of topology, knot theory is an active area of mathematical investigation that has strong connections with a diverse set of scientific fields such as algebra, biology, and statistical mechanics. A popular and important concept in linear algebra, Gram determinants enjoy a connection with the mathematical theory of knots. In this article, we expose this concept and present several types of Gram determinants in what can be considered as a survey of the current Gram determinants of interest to knot theorists; examples are included to illustrate the definitions. In particular, we pay special attention to a recently defined determinant from a Möbius band and we further study its structure. At the end, some speculation is presented regarding the closed formula for the Gram determinant of type $(Mb)_1$, a problem that arouses serious interest among knot theorists. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09704 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Study of Gram Determinants in Knot Theory Ibarra, Dionne Montoya-Vega, Gabriel Geometric Topology Primary: 57M25 Secondary: 57M27 Historically originated as a sub-field of topology, knot theory is an active area of mathematical investigation that has strong connections with a diverse set of scientific fields such as algebra, biology, and statistical mechanics. A popular and important concept in linear algebra, Gram determinants enjoy a connection with the mathematical theory of knots. In this article, we expose this concept and present several types of Gram determinants in what can be considered as a survey of the current Gram determinants of interest to knot theorists; examples are included to illustrate the definitions. In particular, we pay special attention to a recently defined determinant from a Möbius band and we further study its structure. At the end, some speculation is presented regarding the closed formula for the Gram determinant of type $(Mb)_1$, a problem that arouses serious interest among knot theorists. |
| title | A Study of Gram Determinants in Knot Theory |
| topic | Geometric Topology Primary: 57M25 Secondary: 57M27 |
| url | https://arxiv.org/abs/2402.09704 |