Shuffle theorem for torus link homology
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866918325754265600 |
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| author | Kim, Donghyun Oh, Jaeseong |
| author_facet | Kim, Donghyun Oh, Jaeseong |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_06338 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Shuffle theorem for torus link homology Kim, Donghyun Oh, Jaeseong Combinatorics Geometric Topology Quantum Algebra Representation Theory 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. |
| title | Shuffle theorem for torus link homology |
| topic | Combinatorics Geometric Topology Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2602.06338 |