Discrete Koenigs nets and finite Laplace sequences

TL;DR

This paper investigates discrete Koenigs nets, a special class of Q-nets, demonstrating that if their Laplace sequences terminate in one direction, they are finite, revealing new properties of these geometric structures.

Contribution

It establishes a connection between degeneracy in Laplace sequences and the finiteness of these sequences for discrete Koenigs nets.

Findings

Laplace sequences terminate in special cases of Koenigs nets.

Degeneracy after m steps in one direction implies degeneracy after m+1 or m+2 steps in the other.

Finite Laplace sequences are characterized by degeneracy conditions.

Abstract

Q-nets are maps from the square grid to projective space that have planar faces. We consider the Laplace sequences of Q-nets, which are determined by iterating a discrete time dynamics called Laplace transformations. In general, the Laplace sequences are bi-infinite. However, there are special cases in which a Laplace transform degenerates to a curve. In these cases we say that the sequences terminates. In this paper, we consider two special cases of Q-nets which are both called (discrete) Koenigs nets. For these Koenigs nets we show that if the sequence terminates, then the sequence is finite. More specifically, we show that if the Laplace transform is Laplace degenerate (or Goursat degenerate) after m steps in one direction, then it is Laplace degenerate after m + 1 (or m + 2) steps in the other direction.