Thurston'ın ardından küre üçgenlemeleri

Thurston proved that space of non-negatively curved triangulations of the sphere can be parametrized by a quotient of positive part of a Lorentzian lattice of rank 10 by a group of isomorphisms of the lattice. Also square-norm of a lattice point gives number of triangles in corresponding triangulation. We extended his result to 2 new families of triangulations. More precisely we prove that there are 2 Eisenstein lattices with a Hermitian product on them such that positive part of each lattice divided by some isomorphism group of the lattice parametrizes space of triangulations with 4 singular points incident to 3 triangles and space of triangulation with 6 singular points incident to 4 triangles, respectively. Also we prove that square-norm of a lattice point gives number of triangles in corresponding triangulation. In addition we prove a similar result for space of quadrangulations having 4 singular points incident to 2 quadrangles. Our approach gives another proof of Thurston's theorem and Ayberk Zeytin's theorem which is an analogue of Thurston's result for non-negatively curved quadrangulations.