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We introduce a notion of complexity for Sefiert homology spheres byestablishing a correspondence between lattice point counting in tethrahedra andthe Heegaard-Floer homology. This complexity turns out to be equivalent to aversion of Casson invariant and it is monotone under a natural partial order inthe set of Seifert homology spheres. Using this interpretation we prove thatthere are finitely many Seifert homology spheres with prescribed Heegaard-Floerhomology. As an application, we characterize L-spaces and weakly ellipticmanifolds among Seifert homology spheres. Also, we list all the Seiferthomology spheres up to complexity two.

Calculating Heegaard-Floer Homology by Counting Lattice Points in Tetrahedra