  We begin with modular form periods, a focal point of several Yuri Manin's
works. The similarity between the corresponding zeta-polynomials and
superpolynomials in the theory of refined knot invariants is discussed. We
present 3 constructions (conjecturally coinciding) of superpolynomials: via
DAHA, compactified Jacobians and $L$-functions of plane curve singularities and
provide some super analogs of $\rho_{ab}-$ invariants (which is new). They
conjecturally satisfy the Riemann Hypothesis in some sector of the parameters.
Presumably, they can be interpreted as partition functions of certain
Landau-Ginzburg models, and there is a remarkable similarity with the Lee-Yang
theorem. General perspectives of the passage to isolated curve and surface
singularities are discussed, including possible implications in number theory.

Quantum Algebra;Number Theory;Representation Theory