On unstabilized genus three critical Heegaard surfaces

摘要

In this article, we give a sufficient condition for a Heegaard splitting of non-splittable, tunnel number two, three component link exterior to be critical. Moreover, we prove that if F is a genus three, unstabilized, topologically minimal Heegaard surface of a closed, irreducible 3-manifold and it is not an amalgamation of two genus 2 Heegaard surfaces along a torus, then the topological index of F is at most two.