  The Bollob\'as-Riordan polynomial [Math. Ann. 323, 81 (2002)] is a universal
polynomial invariant for ribbon graphs. We find an extension of this polynomial
for a particular family of combinatorial objects, called rank 3 weakly-colored
stranded graphs. Stranded graphs arise in the study of tensor models for
quantum gravity in physics, and generalize graphs and ribbon graphs. We present
a seven-variable polynomial invariant of these graphs, which obeys a
contraction/deletion recursion relation similar to that of the Tutte and
Bollob\'as-Riordan polynomials. However, it is defined on a much broader class
of objects, and furthermore captures properties that are not encoded by the
Tutte or Bollob\'as-Riordan polynomials.
