Combinatorial properties of the K3 surface: Simplicial blowups and slicings

TL;DR

This paper presents a minimal vertex triangulation of the K3 surface by resolving singularities of a 4-dimensional Kummer variety through simplicial blowups, and explores its topological slicings using Morse functions.

Contribution

It introduces a novel minimal 17-vertex triangulation of the K3 surface via explicit simplicial blowups of the Kummer variety.

Findings

Constructed a 17-vertex triangulation of the K3 surface.

Developed a triangulated mapping cylinder of the Hopf map with minimal vertices.

Analyzed topological slicings of the K3 surface using simplicial Morse functions.

Abstract

The 4-dimensional abstract Kummer variety K^4 with 16 nodes leads to the K3 surface by resolving the 16 singularities. Here we present a simplicial realization of this minimal resolution. Starting with a minimal 16-vertex triangulation of K^4 we resolve its 16 isolated singularities - step by step - by simplicial blowups. As a result we obtain a 17-vertex triangulation of the standard PL K3 surface. A key step is the construction of a triangulated version of the mapping cylinder of the Hopf map from the real projective 3-space onto the 2-sphere with the minimum number of vertices. Moreover we study simplicial Morse functions and the changes of their levels between the critical points. In this way we obtain slicings through the K3 surface of various topological types.