2025/07/24 by Burke, Rhuaidi
paper · doi:10.20382/jocg.v16i2a4
Dimension 4 is the first dimension in which exotic smooth manifold pairs appear—manifolds which are topologically the same but for which there is no smooth deformation of one into the other. Whilst smooth and piecewise-linear structures on 4-manifolds do coincide, comparatively little work has been done towards gaining an understanding of smooth 4-manifolds from the discrete and algorithmic perspective. In this paper we introduce new software tools to make this possible, including a software implementation of an algorithm which enables us to build triangulations of 4-manifolds from suitable Kirby diagrams, as well as a new heuristic for simplifying 4-manifold triangulations. Using these tools, we present new triangulations of several exotic pairs with boundary, corks and plugs (objects responsible for "exoticity"), as well as the smallest known triangulation of the K3 surface. The small size of these triangulations benefits us by revealing fine structural features in 4-manifold triangulations.