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A Study of Gram Determinants in Knot Theory

Dionne Ibarra, Gabriel Montoya-Vega

TL;DR

This work analyzes Gram determinants arising from relative Kauffman bracket skein modules in knot theory across multiple surface settings (disk, annulus, Möbius band). It defines bilinear forms on diagram bases, builds corresponding Gram matrices, and derives (or conjectures) closed-form expressions and factorization patterns using tools like Chebyshev polynomials and Jones–Wenzl idempotents. Key contributions include explicit determinant formulas for Type A and Type B, structured divisibility results for Type Mb, and a new (Mb)_1 construction with connections to existing determinants. The results illuminate how linear-algebraic invariants interact with skein-theoretic and topological structures, with potential implications for knot invariants and 3-manifold invariants.

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.

A Study of Gram Determinants in Knot Theory

TL;DR

This work analyzes Gram determinants arising from relative Kauffman bracket skein modules in knot theory across multiple surface settings (disk, annulus, Möbius band). It defines bilinear forms on diagram bases, builds corresponding Gram matrices, and derives (or conjectures) closed-form expressions and factorization patterns using tools like Chebyshev polynomials and Jones–Wenzl idempotents. Key contributions include explicit determinant formulas for Type A and Type B, structured divisibility results for Type Mb, and a new (Mb)_1 construction with connections to existing determinants. The results illuminate how linear-algebraic invariants interact with skein-theoretic and topological structures, with potential implications for knot invariants and 3-manifold invariants.

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 , a problem that arouses serious interest among knot theorists.
Paper Structure (6 sections, 9 theorems, 55 equations, 13 figures, 6 tables)

This paper contains 6 sections, 9 theorems, 55 equations, 13 figures, 6 tables.

Table of Contents

  1. Introduction to Gram Determinants in Knot Theory
  2. Determinant of Type A
  3. Generalization of Type A
  4. Determinant of Type B
  5. Determinant of Type Mb
  6. Determinant of Type $(Mb)_1$

Key Result

Theorem 1.4

KauffmanBracket The diagrammatic algebra is isomorphic to $TL_n$ and can be thought of as a diagrammatic interpretation of it.

Figures (13)

  • Figure 1: The skein triple.
  • Figure 2: A graphical interpretation of the generators of $\mathit{TL}_n$.
  • Figure 3: An illustration of multiplication.
  • Figure 4: The bilinear form on two elements in $\mathit{TL}_4$. Here, the result is $d^2$.
  • Figure 5: The bilinear form of generalized type $A$.
  • ...and 8 more figures

Theorems & Definitions (30)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 2.1
  • Example 2.2
  • Theorem 2.3
  • Definition 3.1
  • Example 3.2
  • Definition 4.1
  • ...and 20 more