The block number $Bl(M)$ introduced in our previous paper is a new topological invariant of a closed orientable 3-manifold $M$ which estimates a combinatorial complexity of $M$ just like the Heegaard genus $HG(M)$. In our previous paper, we have shown an inequality $HG(M) \leq Bl(M)$ for any $M \ne S^2 \times S^1$. In this paper, we will show that $Bl(M)=HG(M)$ for any $M$ with $HG(M)=2$ and moreover that $Bl(M) \leq 4$ for any $M$ with $HG(M)=3$.
"On DS-diagrams for 3-manifolds of Heegaard Genus 2." Tokyo J. Math. 29 (1) 117 - 146, June 2006. https://doi.org/10.3836/tjm/1166661871