A geometric construction of Coxeter-Dynkin diagrams of bimodal singularities

TL;DR

This paper constructs a triangulated category linked to bimodal singularities, using geometric methods to relate Grothendieck groups and Coxeter-Dynkin diagrams, enhancing understanding of singularity classifications.

Contribution

It introduces a geometric construction connecting bimodal singularities with Coxeter-Dynkin diagrams via triangulated categories and Grothendieck groups.

Findings

Grothendieck group described by Coxeter-Dynkin diagram

Construction of a triangulated category for bimodal singularities

Relation between vanishing cycles and diagram basis

Abstract

We consider the Berglund-H\"ubsch transpose of a bimodal invertible polynomial and construct a triangulated category associated to the compactification of a suitable deformation of the singularity. This is done in such a way that the corresponding Grothendieck group with the (negative) Euler form can be described by a graph which corresponds to the Coxeter-Dynkin diagram with respect to a distinguished basis of vanishing cycles of the bimodal singularity.