Fukaya categories have interesting moduli spaces of objects. We study some Fukaya categories involving Lagrangian surfaces bounding Legendrian knots, and others involving Lagrangian three-folds bounding Legendrian surfaces. Both cases can be described explicitly with planar graphs, and both cases relate to cluster theory. We exploit this structure to make enumerative predictions about Lagrangian threefolds. In the interest of time and pedagogy, I will focus on examples.

This talk is a summary of several joint works with subsets of Shen, Shende, Treumann, and Williams, and incorporates prior works of many other researchers, including: Fock-Goncharov, Cecotti-Cordova-Vafa, Goncharov-Kenyon, Dimofte-Gabella-Goncharov, Aganagic-Klemm-Vafa, and Kontsevich-Soibelman.

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