Negative Finding for the Three-Dimensional Dimer Problem
J. M. Hammersley, A. Feuerverger, Alan Julian Izenman, K. Makani · Journal of Mathematical Physics · 1969
The dimer problem can be solved if one can evaluate the permanent of P = (pij), the incidence matrix of the lattice. All known methods of solving the two-dimensional case consist (explicitly or implicitly) in finding another matrix Q = (qij), such that pij = |qij| and per P = |det Q|, and then computing the determinant of Q. We show that in the three-dimensional case no such matrix Q exists for any choice of elements qij, whether real or complex numbers, or quaternions. A stronger negative result of an asymptotic character seems to be true, but this rests upon a plausible but unproved conjecture.