Shuffle theorem for torus link homology

TL;DR

This work establishes a torus-link analogue of the shuffle theorem by proving that the elliptic Hall algebra action $e_{(1^k)}[-MX^{m,n}]\cdot 1$ generates the combinatorial data of $k$-tuples of cyclic $(m,n)$-parking functions, thereby linking symmetric-function operators to Khovanov–Rozansky homology of torus links $T(km,kn)$. The authors develop a determinantal, Jacobi–Trudi style framework using new $\mathfrak{h}$-operators and a rational Shareshian–Wachs involution via wiggle diagrams, reformulating the rational shuffle theorem in terms of $P$-tableaux and a statistic-preserving bijection to standard parking-function tuples. This approach yields a direct combinatorial expression for $e_{(1^k)}[-MX^{m,n}]\cdot 1$ as a generating function over cyclic parking functions and provides elementary proofs of Loehr–Warrington type formulas for $\nabla^m s_{\lambda}$, thereby enriching the interplay between elliptic Hall algebras, diagonal coinvariants, and knot homology. The results give a rigorous spine for the knot-theoretic conjectures of GMV17 and Wilson in the torus-link case and illuminate how new determinantal operator methods can drive positive Laurant expansions in this highly structured setting.

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.

Figures (11)

• Figure 1: An example of an element of $\mathop{\mathrm{PF}}\nolimits^{2}_{4,3}$ (labels are specified in the context).
• Figure 2: The corner, overlapped pairs, adjacent pairs, and the crossing.
• Figure 3: A bigon and a simple region inside it.
• Figure 4: Wiggle diagram $\Pi$ and its $\mathop{\mathrm{SW}}\nolimits(\Pi)$.
• Figure 5: Applications of operations (I) and (II).

Theorems & Definitions (60)

• Theorem 1.1
• Corollary 1.3
• Proposition 2.1
• Lemma 2.2
• Remark 2.3
• Example 2.4
• Definition 3.1
• Lemma 3.2
• proof
• Theorem 3.3