A Study of Gram Determinants in Knot Theory

Dionne Ibarra; Gabriel Montoya-Vega

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arXiv · 2402.09704

A Study of Gram Determinants in Knot Theory

Dionne Ibarra·Gabriel Montoya-Vega

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Abstract

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.

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Dionne Ibarra, Gabriel Montoya-Vega. 2025-11-27. A Study of Gram Determinants in Knot Theory. https://arxiv.org/abs/2402.09704

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