Cutting arcs for torus links and trees

TL;DR

This paper characterizes certain torus links and Coxeter-Dynkin trees based on the finiteness of cutting arcs that preserve fibredness, linking geometric properties with algebraic classifications.

Contribution

It provides a novel characterization of torus links from simple plane curve singularities and Coxeter-Dynkin trees using the concept of cutting arcs that preserve fibredness.

Findings

Characterization of torus links with finite fibred-preserving cutting arcs

Identification of Coxeter-Dynkin trees via similar cutting arc properties

Connection between geometric link properties and algebraic classifications

Abstract

Among all torus links, we characterise those arising as links of simple plane curve singularities by the property that their fibre surfaces admit only a finite number of cutting arcs that preserve fibredness. The same property allows a characterisation of Coxeter-Dynkin trees (i.e., $A_n$, $D_n$, $E_6$, $E_7$ and $E_8$) among all positive tree-like Hopf plumbings.