Are there $p$-adic knot invariants?

We suggest using the Hall–Littlewood version of the Rosso–Jones formula to define the germs of $p$-adic HOMFLY-PT polynomials for torus knots $[m,n]$ as coefficients of superpolynomials in a $q$-expansion. In this form, they have at least the $[m,n]\leftrightarrow[n,m]$ topological invariance. This opens a new possibility to interpret superpolynomials as $p$-adic deformations of HOMFLY polynomials and poses a question of generalizing to other knot families, which is a substantial problem for several branches of modern theory.