A symplectic basis for 3-manifold triangulations

TL;DR

This paper provides a geometric interpretation of Neumann and Zagier's symplectic structure for 3-manifold triangulations, linking it to topology via curve intersections on a Heegaard surface, and offers an algorithm to construct a symplectic basis.

Contribution

It introduces a topological interpretation of the symplectic structure and presents an algorithm for constructing a symplectic basis for 3-manifold triangulations.

Findings

Geometric interpretation of the symplectic structure in terms of curve intersections.

Algorithm for constructing a symplectic basis from curves.

Connection between symplectic structures and 3-manifold topology.

Abstract

In the 1980s, Neumann and Zagier introduced a symplectic vector space associated to an ideal triangulation of a cusped 3-manifold, such as a knot complement. We give a geometric interpretation for this symplectic structure in terms of the topology of the 3-manifold, via intersections of certain curves on a Heegaard surface. We also give an algorithm to construct curves forming a symplectic basis.