Lotuses as computational architectures

Lotuses are certain types of finite contractible simplicial complexes, obtained by identifying vertices of polygons subdivided by diagonals. As we explained in a previous paper, each time one resolves a complex reduced plane curve singularity by a sequence of toroidal modifications with respect to suitable local coordinates, one gets a naturally associated lotus, which allows one to unify the classical trees used to encode the combinatorial type of the singularity. In this paper we explain how to associate a lotus to each constellation of crosses , which is a finite constellation of infinitely near points endowed with compatible germs of normal crossings divisors with two components, and how this lotus may be seen as a computational architecture . Namely, if the constellation of crosses is associated to an embedded resolution of a complex reduced plane curve singularity A A , one may compute progressively as vertex and edge weights on the lotus the log-discrepancies of the exceptional divisors, the orders of vanishing on them of the starting coordinates, the multiplicities of the strict transforms of the branches of A A , the orders of vanishing of a defining function of A A , the associated Eggers-Wall tree, the delta invariant and the Milnor number of A A , etc. We illustrate these computations using three recurrent examples. Finally, we describe the changes to be done when one works in positive characteristic.