Mathematical Research Letters

Volume 29 (2022)

Number 6

Diagrams of $\star$-trisections

Pages: 1595 – 1658



Román Aranda (Department of Mathematics and Statistics, Binghamton University, Binghamton, New York, U.S.A.)

Jesse Moeller (Mathematics, University of Nebraska, Lincoln, Ne., U.S.A.)


In this note we provide a generalization for the definition of a trisection of a $4$-manifold with boundary. We demonstrate the utility of this more general definition by finding a trisection diagram for the Cacime Surface, and also by finding a trisection-theoretic way to perform logarithmic surgery. In addition, we describe how to perform $1$-surgery on closed trisections. The insight gained from this description leads us to the classification of an infinite family of genus three trisections. We include an appendix where we extend two classic results for relative trisections for the case when the trisection surface is closed.

Received 11 December 2019

Accepted 23 November 2020

Published 4 May 2023