In a companion paper (arXiv 2011.01768) we constructed non-negative integer
coordinates Ξ¦Tβ for a distinguished collection W3,Sβ of SL3β-webs on a finite-type punctured surface
S, depending on an ideal triangulation T of
S. We prove that these coordinates are natural with respect to the
choice of triangulation, in the sense that if a different triangulation
Tβ² is chosen then the coordinate change map relating
Ξ¦Tβ and Ξ¦Tβ²β is a prescribed tropical
cluster transformation. Moreover, when S=β‘ is an ideal square,
we provide a topological geometric description of the Hilbert basis (in the
sense of linear programming) of the non-negative integer cone
Ξ¦Tβ(W3,β‘β)βZβ₯012β, and
we prove that this cone canonically decomposes into 42 sectors corresponding
topologically to 42 families of SL3β-webs in the square.Comment: 39 pages, 27 figure