Heegaard splittings and the pants complex
  We define integral measures of complexity for Heegaard splittings based on the graph dual to the curve complex and on the pants complex defined by Hatcher and Thurston. As the Heegaard splitting is stabilized, the sequence of complexities turns out to converge to a non-trivial limit depending only on the manifold. We then use a similar method to compare different manifolds, defining a distance which converges under stabilization to an integer related to Dehn surgeries between the two manifolds. 
Jesse Johnson