Tropical complexes of submatrices of tree metrics

PASS

The proposed counterexample is mathematically valid for the natural interpretation that the split system is the set of edge splits of $C_S$. The two configurations in $\mathbb{TT}^{\{1,2,3\}}$ have the same edge-direction split set $\{1|23,2|13,12|3\}$, but their tropical complexes have different dimensions: $C_{S_1}$ is a tripod, while $C_{S_2}$ has two-dimensional cells. Hence the split system alone cannot determine the combinatorics of $S$ or $C_S$.

Novelty assessment

KNOWN

Classification rationale: The accepted counterexample is not a new resolution. Under the yes/no interpretation “does the split system determine the full combinatorics?”, the negative answer is already implicit in standard tropical-plane examples and in Wu–Xu’s own paper: in $\mathbb{TT}^3$, the split system records only which of the three possible edge directions occur, so 1-dimensional and 2-dimensional tropical complexes can have the same split set.

Literature check: The original Wu–Xu paper defines edge splits just before Example 4.8. Example 4.9 then exhibits, for a three-point tree metric, a 2-dimensional tropical complex $C_D$ and a 1-dimensional star $-\operatorname{tconv}[-D]$; both have edges in the same three split directions of $X=\{a,b,c\}$, but their face posets/dimensions differ. This is essentially the same phenomenon as the proposed $S_1,S_2$ counterexample. Searches for the exact question and related phrases did not reveal a later standalone “solution”; the relevant counterexample is already present in the source literature.

Citation: Y. Wu and Z. Xu, “Tropical complexes of submatrices of tree metrics,” 2017, definition before Example 4.8 and Example 4.9. See also Develin–Sturmfels, “Tropical Convexity,” Documenta Math. 9 (2004), for the basic tropical-complex framework.