Contact topology and holomorphic invariants via elementary combinatorics

In recent times a great amount of progress has been achieved in symplectic and contact geometry, leading to the development of powerful invariants of 3-manifolds such as Heegaard Floer homology and embedded contact homology. These invariants are based on holomorphic curves and moduli spaces, but in the simplest cases, some of their structure reduces to some elementary combinatorics and algebra which may be of interest in its own right. In this note, which is essentially a light-hearted exposition of some previous work of the author, we give a brief introduction to some of the ideas of contact topology and holomorphic curves, discuss some of these elementary results, and indicate how they arise from holomorphic invariants.