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For a lattice path $\nu$ from the origin to a point $(a,b)$ using steps$E=(1,0)$ and $N=(0,1)$, we construct an associated flow polytope$\mathcal{F}_{\hat{G}_B(\nu)}$ arising from an acyclic graph wherebidirectional edges are permitted. We show that the flow polytope$\mathcal{F}_{\hat{G}_B(\nu)}$ admits a subdivision dual to a $w$-simplex,where $w$ is the number of valleys in the path $\bar{\nu} = E\nu N$.Refinements of this subdivision can be obtained by reductions of a polynomial$P_\nu$ in a generalization of M\'esz\'aros' subdivision algebra for acyclicroot polytopes where negative roots are allowed. Via an integral equivalencebetween $\mathcal{F}_{\hat{G}_B(\nu)}$ and the product of simplices$\Delta_a\times \Delta_b$, we thereby obtain a subdivision algebra for aproduct of two simplices. As a special case, we give a reduction order forreducing $P_\nu$ that yields the cyclic $\nu$-Tamari complex of Ceballos,Padrol, and Sarmiento.

A subdivision algebra for a product of two simplices via flow polytopes