Loho, Georg PeterSmith, Ben2018-10-042018-10-042018-10-04202010.1016/j.aim.2020.107232https://infoscience.epfl.ch/handle/20.500.14299/1486541804.01595We show that the Chow covectors of a linkage matching field define a bijection of lattice points and we demonstrate how one can recover the linkage matching field from this bijection. This resolves two open questions from Sturmfels & Zelevinsky (1993) on linkage matching fields. For this, we give an explicit construction that associates a bipartite incidence graph of an ordered partition of a common set to all lattice points in a dilated simplex. Given a triangulation of a product of two simplices encoded by a set of bipartite trees, we similarly prove that the bijection from left to right degree vectors of the trees is enough to recover the triangulation. As additional results, we show a cryptomorphic description of linkage matching fields and characterise the flip graph of a linkage matching field in terms of its prodsimplicial flag complex. Finally, we relate our findings to transversal matroids through the tropical Stiefel map.Matching fieldtriangulationlattice pointsproduct of simpliceslinkage propertybipartite graphtext::journal::journal article::research article