On classification of Heegaard splittings and triangulations

In this paper we consider Heegaard splittings of 3-manifolds. By using Gabai’s concept of thin position on the 1-skeleton of some polyhedral decomposition, together with CassonGordon’s concept of strong irreducibility, we prove the Main Theorem (4.0). This theorem will allow us to classify the Heegaard splittings of manifolds whose polyhedral decompositions are particularily nice, which we demonstrate via examples. Specically, we use it to classify Heegaard splittings of several hyperbolic spaces, including the gure-8 knot complement (Example 6.4) and the genus 2 case of the 52-knot complement (Example 6.7).