Contact topology and holomorphic invariants via elementary combinatorics

TL;DR

This paper provides an accessible overview of contact topology and holomorphic invariants, highlighting how elementary combinatorics and algebra relate to complex geometric structures in symplectic and contact geometry.

Contribution

It introduces simplified combinatorial and algebraic perspectives on holomorphic invariants in contact topology, connecting elementary results to advanced geometric concepts.

Findings

Elementary combinatorics can describe aspects of holomorphic invariants.

Simplified algebraic structures relate to complex geometric invariants.

The approach offers a pedagogical perspective on contact topology.

Abstract

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.