Splice diagrams and splice-quotient surface singularities

TL;DR

This paper reviews the development and application of splice diagrams in classifying certain surface singularities, discussing key conjectures and open questions in the field.

Contribution

It introduces splice diagrams for defining splice type and splice-quotient singularities, and discusses related conjectures and open problems in surface singularity theory.

Findings

Summary of results and methods from collaboration with Walter Neumann

Discussion of the Casson Invariant Conjecture and Milnor Fiber Conjectures

Open questions about hypersurface and complete intersection singularities

Abstract

The current work will appear in a Celebratio Mathematica volume in honor of Walter Neumann. We summarize results and methods from our long-time collaboration with Neumann, especially the motivation for the introduction of splice diagrams to define singularities of splice type and splice-quotient singularities. The Casson Invariant Conjecture and Milnor Fiber Conjectures are discussed, as well as some open questions about hypersurface and complete intersection singularities with integral homology sphere link.