Tall Complexity One Spaces with k-colorable Skeleton

Tall complexity one $T$-spaces are Hamiltonian $T$-spaces $(M,\omega,\Phi)$ such that $\frac{1}{2}\dim M -\dim T=1$ and the symplectic quotient at each moment value is a surface. The skeleton of a complexity one $T$-space is an important invariant in the classification and encodes the information about non-generic orbits. In this paper, we study properties of the skeleton of a compact, connected tall complexity one $T$-spaces. We prove that when the skeleton is $k$-colorable, i.e., when it can be partitioned into $k$ closed and open subsets such that the orbital moment map is injective on each of them, its information can be recovered by the one-skeleton (the set of non-generic orbits whose dimension is at most one). We also prove that for any cloesd and open subset of the skeleton on which the orbital moment map is injective, one can construct a symplectic toric $(T\times S^1)$-manifold whose underlying complexity one $T$-space has the skeleton isomorphic to this subset.