Knot Invariants from Laplacian Matrices

TL;DR

This paper introduces a novel method to derive knot invariants such as Seifert, Alexander, and signature matrices from Laplacian matrices of specially constructed graphs associated with link diagrams, unifying graph theory and knot theory.

Contribution

It presents a new approach to compute classical knot invariants using Laplacian matrices of directed, weighted graphs derived from link diagrams, extending previous methods with a device for general diagrams.

Findings

Principal minors of Laplacian matrices yield Seifert and Alexander matrices.

Method applies to general diagrams via Kauffman's device.

Provides a unified graph-theoretic framework for knot invariants.

Abstract

A checkerboard graph of a special diagram of an oriented link is made a directed, edge-weighted graph in a natural way so that a principal minor of its Laplacian matrix is a Seifert matrix of the link. Doubling and weighting the edges of the graph produces a second Laplacian matrix such that a principal minor is an Alexander matrix of the link. The Goeritz matrix and signature invariants are obtained in a similar way. A device introduced by L. Kauffman makes it possible to apply the method to general diagrams.