Gorsky–Oblomkov–Rasmussen conjecture for stable Khovanov homology

From papers

For n1n\geq 1, let T(n,)T(n,\infty) denote the stable limit of the shifted Khovanov homologies of the nn-strand torus links. Let s0,,sn1Z[x0,,xn1]s_0,\dots,s_{n-1}\in\mathbb Z[x_0,\dots,x_{n-1}] be given by

sk=i=0kxixki.s_k=\sum_{i=0}^k x_i x_{k-i}.

Give the variables the bigradings h(xi)=2ih(x_i)=2i and q(xi)=2i+2q(x_i)=2i+2.

Gorsky–Oblomkov–Rasmussen conjecture. The stable Khovanov homology Kh(T(n,))\operatorname{Kh}(T(n,\infty)) is the dual to the homology of the Koszul complex associated to the sequence s0,,sn1s_0,\dots,s_{n-1}, with the Koszul differential preserving quantum grading and lowering homological grading by 11.

This conjecture gives an algebraic model for stable Khovanov homology of torus links. The supplied text introduces the claim but does not state whether it has been proved or disproved.

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Sources & referencesView supporting material

Primary source

Tuomas Kelomäki, “Morse matchings and Khovanov homology of 4-strand torus links”, arXiv:2507.15060 (2025).

Additional references

4 papers in this index state this conjecture (2017–2025). The statement above is taken from the most recent of them; the others are arXiv:2402.10332, arXiv:2109.12889, arXiv:1706.08919.

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