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In this paper we introduce a representation of a embedded knotted (sometimesLagrangian) tori in $\BR^4$ called a hypercube diagram, i.e., a 4-dimensionalcube diagram. We prove the existence of hypercube homology that is invariantunder 4-dimensional cube diagram moves, a homology that is based on knot Floerhomology. We provide examples of hypercube diagrams and hypercube homology,including using the new invariant to distinguish (up to cube moves) two "Hopflinked" tori. We also give examples of a "Trefoil" torus and an immersedknotted torus that is an amalgamation of the $5_2$ knot and a trefoil knot.

Embedded and Lagrangian Knotted Tori in $\BR^4$ and Hypercube Homology