Superpolynomials of algebraic links

TL;DR

This paper develops the theory of motivic superpolynomials for algebraic links, exploring their relations to $L$-functions, DAHA, and applications in algebraic geometry and knot theory, proposing conjectures and generalizations.

Contribution

It introduces a comprehensive framework connecting motivic superpolynomials with DAHA, $L$-functions, and algebraic links, extending existing conjectures and applications.

Findings

Conjectural coincidence of motivic superpolynomials with DAHA superpolynomials.

Relations established between superpolynomials and $L$-functions of plane curve singularities.

Applications to affine Springer fibers, Jacobians, and algebraic knots.

Abstract

Theory of motivic superpolynomials is developed, including its extension to algebraic links colored by rows, relations to $L$-functions of plane curve singularities, the justification of the motivic versions of Weak Riemann Hypothesis, and recurrences for iterated torus links. The key theme is the conjectural coincidence of motivic superpolynomials with the DAHA ones, which can be interpreted as a far-reaching generalization of the Shuffle Conjecture. Applications include affine Springer fibers of type $A_n$ and compactified Jacobians in the most general case (for arbitrary characteristic polynomials) and extended rho-invariants of algebraic knots. The 2nd connection conjecture relates the superpolynomials to the Galkin-St\"ohr $L$-functions, which is some counterpart of the ORS conjecture. The corresponding theory of plane curve singularities is systematically exposed and developed, which can be seen in the case of Hopf links as a generalized version of Schubert Calculus.