The Kalmanson Complex
Abstract
Let X be a finite set of cardinality n. The Kalmanson complex Kn 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 Kn from three perspectives. In addition to the T-theoretic description, we show that Kn 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 Kn.
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