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We define a new topological polynomial extending the Bollobas-Riordan one,which obeys a four-term reduction relation of the deletion/contraction type andhas a natural behavior under partial duality. This allows to write down acompletely explicit combinatorial evaluation of the polynomials, occurring inthe parametric representation of the non-commutative Grosse-Wulkenhaar quantumfield theory. An explicit solution of the parametric representation forcommutative field theories based on the Mehler kernel is also provided.

Topological graph polynomials and quantum field theory, Part II: Mehler kernel theories