#### Vol. 294, No. 2, 2018

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Diagrams for relative trisections

### Nickolas A. Castro, David T. Gay and Juanita Pinzón-Caicedo

Vol. 294 (2018), No. 2, 275–305
##### Abstract

We establish a correspondence between trisections of smooth, compact, oriented $4$-manifolds with connected boundary and diagrams describing these trisected $4$-manifolds. Such a diagram comes in the form of a compact, oriented surface with boundary together with three tuples of simple closed curves, with possibly fewer curves than the genus of the surface, satisfying a pairwise condition of being standard. This should be thought of as the $4$-dimensional analog of a sutured Heegaard diagram for a sutured $3$-manifold. We also give many foundational examples.

##### Keywords
trisection, open book, Heegaard diagram, monodromy, 3-manifold, 4-manifold
Primary: 57M99
##### Milestones
Received: 18 January 2017
Revised: 16 October 2017
Accepted: 30 October 2017
Published: 20 February 2018
##### Authors
 Nickolas A. Castro Department of Mathematics University of California, Davis One Shields Ave Davis, CA 95616 United States David T. Gay Euclid Lab 160 Milledge Terrace Athens, GA 30606 United States Juanita Pinzón-Caicedo Department of Mathematics NC State University 2108 SAS Hall Raleigh, NC 27695 United States