Checkerboard graph links and simply laced Dynkin diagrams

TL;DR

This paper establishes a connection between signed graphs with eigenvalues greater than -2 and simply laced Dynkin diagrams, and applies this to classify certain fibred links with maximal signature.

Contribution

It introduces an equivalence relation on signed graphs linking their adjacency matrices to Dynkin diagrams and classifies checkerboard graph links with maximal signature.

Findings

Signed graphs with eigenvalues > -2 are equivalent to Dynkin diagrams.

Checkerboard graph links with maximal signature correspond to Dynkin diagram links.

The classification includes all such links as isotopic to those from Dynkin diagrams.

Abstract

We define an equivalence relation on graphs with signed edges, such that the associated adjacency matrices of two equivalent graphs are congruent over $\mathbb{Z}$. We show that signed graphs whose eigenvalues are larger than $-2$ are equivalent to one of the simply laced Dynkin diagrams: $A_{n}$, $D_{n}$, $E_{6}$, $E_{7}$ and $E_{8}$. Checkerboard graph links are a class of fibred strongly quasipositive links which include positive braid links. We use the previous result to prove that a checkerboard graph link with maximal signature is isotopic to one of the links realized by the simply laced Dynkin diagrams.