We give a purely combinatorial formula for evaluating closed, decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral, equivariant version of colored Khovanov–Rozansky link homology categorifying the \mathfrak {sl}_N link polynomial. We also provide connections to the equivariant cohomology rings of partial flag varieties.

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We provide a finite-dimensional categorification of the symmetric evaluation of sl N -webs using foam technology. As an output we obtain a symmetric link homology theory categorifying the link invariant associated to symmetric powers of the standard representation of sl N . The construction is made in an equivariant setting. We prove also that there is a spectral sequence from the Khovanov-Rozansky triply graded link homology to the symmetric one and provide along the way a foam interpretation of Soergel bimodules.

Box-TDI polyhedra are polyhedra described by systems which yield strong min-max relations. We characterize them in several ways, involving the notions of principal box-integer polyhedra and equimodular matrices.A polyhedron is box-integer if its intersection with any integer box { ≤ x ≤ u} is integer. We define principally box-integer polyhedra to be the polyhedra P such that kP is box-integer whenever kP is integer. A rational r × n matrix is equimodular if it has full row rank and its nonzero r × r determinants all have the same absolute value. A face-defining matrix is a full row rank matrix describing the affine hull of a face of the polyhedron. Our main result is that the following statements are equivalent.• The polyhedron P is box-TDI.• The polyhedron P is principally box-integer.• Every face-defining matrix of P is equimodular.• Every face of P has an equimodular face-defining matrix.• Every face of P has a totally unimodular face-defining matrix.

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