  We describe a construction of the Tutte polynomial for both matroids and
$q$-matroids based on an appropriate partition of the underlying support
lattice into intervals that correspond to prime-free minors, which we call a
Tutte partition. We show that such partitions in the matroid case include the
class of partitions arising in Crapo's definition of the Tutte polynomial,
while not representing a direct $q$-analogue of such partitions. We propose
axioms of $q$-Tutte-Grothendiek invariance and show that this yields a
$q$-analogue of Tutte-Grothendiek invariance. We establish the connection
between the rank polynomial and the Tutte polynomial, showing that one can be
obtained from the other by convolution.
