Difference of mutant knot invariants and their differential expansion

We evaluate the differences of HOMFLY-PT invariants for pairs of mutant knots colored with representations of $SL(N)$, which are large enough to distinguish between them. These mutant pairs include the pretzel mutants, which require at least the representation, labelled by the Young diagram [2, 4]. We discuss the differential expansion for the differences, which is nontrivial in the case of mutants with a nonzero defect. The most effective technical tool in this case turns out to be the standard Reshetikhin-Turaev approach.