From Goeritz matrices to quasi-alternating links

TL;DR

This paper explores the development from Goeritz matrices to the theory of quasi-alternating links, providing historical context and connecting knot theory with related mathematical concepts.

Contribution

It offers a comprehensive overview of the progression from classical matrices to modern quasi-alternating link theory, including historical insights and related topics.

Findings

Historical connection between Kirchhoff and Goeritz matrices

Link determinant relates to electrical network complexity

Framework for understanding quasi-alternating links

Abstract

Knot Theory is currently a very broad field. Even a long survey can only cover a narrow area. Here we concentrate on the path from Goeritz matrices to quasi-alternating links. On the way, we often stray from the main road and tell related stories, especially if they allow as to place the main topic in a historical context. For example, we mention that the Goeritz matrix was preceded by the Kirchhoff matrix of an electrical network. The network complexity extracted from the matrix corresponds to the determinant of a link. We assume basic knowledge of knot theory and graph theory, however, we offer a short introduction under the guise of a historical perspective. Chapter IV of the book "KNOTS: From combinatorics of knot diagrams to combinatorial topology based on knots will be based on this survey,