Shuffle theorem for torus link homology

TL;DR

This paper proves a new mathematical identity linking elliptic Hall algebra functions to torus link homology, resolving longstanding conjectures and establishing algebraic structures governing link invariants.

Contribution

It establishes a novel connection between elliptic Hall algebra and Khovanov--Rozansky homology of torus links, confirming conjectures and advancing understanding of link invariants.

Findings

Proved the shuffle theorem for torus link homology.

Connected elliptic Hall algebra to link homology invariants.

Developed a rational analogue of the Shareshian--Wachs involution.

Abstract

We prove that the symmetric function $e_{(1^k)}[-MX^{m,n}] \cdot 1$, arising from the elliptic Hall algebra, equals the generating function for $k$-tuples of cyclic $(m,n)$-parking functions. This result resolves a conjecture of Gorsky--Mazin--Vazirani and Wilson, establishing that the elliptic Hall algebra governs the Khovanov--Rozansky homology of torus links $T(km,kn)$. Consequently, this provides an affirmative answer to a question of Galashin and Lam in the torus link case. As a key step in the proof, we develop a rational analogue of the Shareshian--Wachs involution originally introduced to prove the symmetry property of the chromatic quasisymmetric functions.