The Kalmanson Complex

TL;DR

The paper explores the Kalmanson complex from multiple perspectives, establishing equivalences among its T-theoretic, geometric, and matrix-based descriptions, and provides insights into its structure and properties.

Contribution

It introduces a unified framework connecting T-theoretic, geometric, and matrix descriptions of the Kalmanson complex, simplifying proofs and advancing understanding of its structure.

Findings

Proves equivalence of three descriptions of K_n

Provides a simplified proof of Kalmanson and circular decomposable metrics equivalence

Partially describes the f-vector of K_n

Abstract

Let X be a finite set of cardinality n. The Kalmanson complex K_n is the simplicial complex whose vertices are non-trivial X-splits, and whose facets are maximal circular split systems over X. In this paper we examine K_n from three perspectives. In addition to the T-theoretic description, we show that K_n has a geometric realization as the Kalmanson conditions on a finite metric. A third description arises in terms of binary matrices which possess the circular ones property. We prove the equivalence of these three definitions. This leads to a simplified proof of the well-known equivalence between Kalmanson and circular decomposable metrics, as well as a partial description of the f-vector of K_n.