Large chirotopes with computable numbers of triangulations

Abstract

Chirotopes are a common combinatorial abstraction of (planar) point sets. In this paper we investigate decomposition methods for chirotopes, and their application to the problem of counting the number of triangulations supported by a given planar point set. In particular, we generalize the convex and concave sums operations defined by Rutschmann and Wettstein for a particular family of chirotopes (which they call chains), and obtain a precise asymptotic estimate for the number of triangulations of the double circle, using a functional equation and the kernel method.

Figures (8)

• Figure 1: Two chains (left), their convex sum (center) and their concave sum (right).
• Figure 2: Example of join of two (realizable) chirotopes.
• Figure 3: Example of meet of two (realizable) chirotopes.
• Figure 4: The bounding boxes.
• Figure 5: Two weak triangulations of the same rooted chirotope $(\chi,u)$. The one on the left is an extension of a triangulation of $\chi$, but not the one on the right.

Theorems & Definitions (39)

• Definition 2.1
• Proposition 2.2
• Corollary 2.3
• proof
• Lemma 2.4
• proof
• Definition 2.5
• Lemma 2.6
• proof : Proof of \ref{['lem:deltaepsilonsqueezing']}
• proof : Proof of \ref{['prop:join_realizable']}