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Knot Theory and Error-Correcting Codes

Altan B. Kilic, Anne Nijsten, Ruud Pellikaan, Alberto Ravagnani

TL;DR

This work builds a rigorous bridge between knot colorings (Fox, Dehn, and Alexander-Briggs) and linear error-correcting codes by using coloring matrices as parity-check matrices to generate knot-derived codes. It analyzes how knot-theoretic data—such as Alexander polynomials, elementary ideals, and diagrammatic operations (Reidemeister moves, connected sums, pretzel/torus constructions)—translate into code parameters like length, dimension, and minimum distance, and it develops LDPC-like code families from alternating diagrams. The paper provides explicit results for two knot families (torus and pretzel), shows how connected sums affect code parameters, and investigates dual-code behavior, offering both constructive coding schemes and open questions on asymptotic performance and LCD properties. Overall, it offers a novel, constructive framework to derive parameter-controlled codes from topological objects, enriching both knot theory and coding theory with new algebraic perspectives and potential decoding approaches.

Abstract

This paper builds a novel bridge between algebraic coding theory and mathematical knot theory, with applications in both directions. We give methods to construct error-correcting codes starting from the colorings of a knot, describing through a series of results how the properties of the knot translate into code parameters. We show that knots can be used to obtain error-correcting codes with prescribed parameters and an efficient decoding algorithm.

Knot Theory and Error-Correcting Codes

TL;DR

This work builds a rigorous bridge between knot colorings (Fox, Dehn, and Alexander-Briggs) and linear error-correcting codes by using coloring matrices as parity-check matrices to generate knot-derived codes. It analyzes how knot-theoretic data—such as Alexander polynomials, elementary ideals, and diagrammatic operations (Reidemeister moves, connected sums, pretzel/torus constructions)—translate into code parameters like length, dimension, and minimum distance, and it develops LDPC-like code families from alternating diagrams. The paper provides explicit results for two knot families (torus and pretzel), shows how connected sums affect code parameters, and investigates dual-code behavior, offering both constructive coding schemes and open questions on asymptotic performance and LCD properties. Overall, it offers a novel, constructive framework to derive parameter-controlled codes from topological objects, enriching both knot theory and coding theory with new algebraic perspectives and potential decoding approaches.

Abstract

This paper builds a novel bridge between algebraic coding theory and mathematical knot theory, with applications in both directions. We give methods to construct error-correcting codes starting from the colorings of a knot, describing through a series of results how the properties of the knot translate into code parameters. We show that knots can be used to obtain error-correcting codes with prescribed parameters and an efficient decoding algorithm.
Paper Structure (17 sections, 53 theorems, 48 equations, 18 figures)

This paper contains 17 sections, 53 theorems, 48 equations, 18 figures.

Table of Contents

  1. Outline.
  2. Knots and Codes
  3. Knot Theory
  4. Coding Theory
  5. Knot Colorings
  6. Fox Coloring
  7. Dehn Coloring
  8. Alexander-Briggs Coloring
  9. Codes from Knots and Their Properties
  10. Dimension of Codes from Knot Diagrams
  11. Two Families of Fox Knot Codes
  12. Torus Knots
  13. Pretzel Knots
  14. Knot Graphs and Their Codes
  15. Connected Sum of Knot Diagrams
  16. ...and 2 more sections

Key Result

Theorem 1.7

Any polygonal knot $K$ is equivalent, under an arbitrarily small rotation of $\mathbb{R}^3$, to a polygonal knot $K'$ for which $p(K')$ is regular.

Figures (18)

  • Figure 1: An example of a trivial and a non-trivial knot.
  • Figure 2: An oriented trefoil knot.
  • Figure 3: The trefoil knot as an entangled polygon and as a torus knot.
  • Figure 4: The Reidemeister moves of type I, II and III, respectively.
  • Figure 5: The figure-eight knot is equivalent to its mirror image.
  • ...and 13 more figures

Theorems & Definitions (165)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.4
  • Example 1.5
  • Definition 1.6
  • Theorem 1.7: see crowell2012introduction
  • Remark 1.8
  • Definition 1.9
  • Lemma 1.10
  • Definition 1.11
  • ...and 155 more