Computing trisections of 4-manifolds

Algorithms that decompose a manifold into simple pieces reveal the geometric and topological structure of the manifold, showing how complicated structures are constructed from simple building blocks. This note describes a way to algorithmically construct a trisection, which describes a $4$-dimensional manifold as a union of three $4$-dimensional handlebodies. The complexity of the $4$-manifold is captured in a collection of curves on a surface, which guide the gluing of the handelbodies. The algorithm begins with a description of a manifold as a union of pentachora, or $4$-dimensional simplices. It transforms this description into a trisection. This results in the first explicit complexity bounds for the trisection genus of a $4$-manifold in terms of the number of pentachora ($4$-simplices) in a triangulation.Comment: 15 pages, 9 figure

On the narrative form of simulations.

Understanding complex physical systems through the use of simulations often takes on a narrative character. That is, scientists using simulations seek an understanding of processes occurring in time by generating them from a dynamic model, thereby producing something like a historical narrative. This paper focuses on simulations of the Diels-Alder reaction, which is widely used in organic chemistry. It calls on several well-known works on historical narrative to draw out the ways in which use of these simulations mirrors aspects of narrative understanding: Gallie for "followability" and "contingency"; Mink for "synoptic judgment"; Ricoeur for "temporal dialectic"; and Hawthorn for a related dialectic of the "actual and the possible". Through these reflections on narrative, the paper aims for a better grasp of the role that temporal development sometimes plays in understanding physical processes and of how considerations of possibility enhance that understanding