Monotone links in DAHA and EHA

TL;DR

This paper introduces monotone links on a torus, connecting them to algebraic structures like DAHA and EHA, and explores their relation to Coxeter and positroid links, conjecturing positivity and homological properties.

Contribution

It defines monotone links on a torus, relates them to algebraic and geometric objects, and extends known conjectures on symmetric functions and link invariants.

Findings

Monotone links correspond to Coxeter links in the flag Hilbert scheme.

Convexity conditions yield positroid links.

Piecewise almost linear convex curves produce algebraic links.

Abstract

We define monotone links on a torus, obtained as projections of curves in the plane whose coordinates are monotone increasing. Using the work of Morton-Samuelson, to each monotone link we associate elements in the double affine Hecke algebra and the elliptic Hall algebra. In the case of torus knots (when the curve is a straight line), we recover symmetric function operators appearing in the rational shuffle conjecture. We show that the class of monotone links viewed as links in $\mathbb R^3$ coincides with the class of Coxeter links, studied by Oblomkov-Rozansky in the setting of the flag Hilbert scheme. When the curve satisfies a convexity condition, we recover positroid links that we previously studied. In the convex case, we conjecture that the associated symmetric functions are Schur positive, extending a recent conjecture of Blasiak-Haiman-Morse-Pun-Seelinger, and we speculate on the relation to Khovanov-Rozansky homology. Our constructions satisfy a skein recurrence where the base case consists of piecewise almost linear curves. We show that convex piecewise almost linear curves give rise to algebraic links.