There is a natural interplay between trisections and Heegaard Floer homology, and in particular with the mixed invariants of Ozsvath and Szabo. We outline how to compute these invariants by gluing together two trisection diagrams and speculate on a possible relationship with Massey products.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/TopologySeminar