The moduli space of n tropically collinear points in R^d

The tropical semiring (R, min, +) has enjoyed a recent renaissance, owing to its connections to mathematical biology as well as optimization and algebraic geometry. In this paper, we investigate the space of labeled n-point configurations lying on a tropical line in d-space, which is interpretable as the space of n-species phylogenetic trees. This is equivalent to the space of d by n matrices of tropical rank two, a simplicial complex. We prove that this simplicial complex is shellable for dimension d=3 and compute its homology in this case, conjecturing that this complex is shellable in general. We also investigate the space of d by n matrices of Barvinok rank two, a subcomplex directly related to optimization, giving a complete description of this subcomplex in the case d=3.