A Tutte-style proof of Brylawski’s tensor product formula

We provide a new proof of Brylawski’s formula for the Tutte polynomial of the tensor product of two matroids. Our proof involves extending Tutte’s formula, expressing the Tutte polynomial using a calculus of activities, to all polynomials involved in Brylawski’s formula. The approach presented here may be used to show a signed generalization of Brylawski’s formula, which may be used to compute the Jones polynomial of some large non-alternating knots. Our proof inspires an extension of Brylawski’s formula to the case when the pointed element is a factor.