[Paper Review] Tautological classes and symmetry in Khovanov-Rozansky homology

Eugene Gorsky, Matthew Hogancamp+1 more|arXiv (Cornell University)|Mar 1, 2021

Homotopy and Cohomology in Algebraic Topology32 references4 citations

TL;DR

This paper introduces a new family of commuting operators $F_k$ in Khovanov-Rozansky link homology, inspired by tautological classes in character variety cohomology. It proves that $F_2$ satisfies the hard Lefshetz property, establishing the symmetry in triply graded homology conjectured by Dunfield, Gukov, and Rasmussen for all knots, using a direct algebraic approach via a dg algebra $\mathcal{A}$ and its coproduct structure.

ABSTRACT

We define a new family of commuting operators $F_k$ in Khovanov-Rozansky link homology, similar to the action of tautological classes in cohomology of character varieties. We prove that $F_2$ satisfies ``hard Lefshetz property" and hence exhibits the symmetry in Khovanov-Rozansky homology conjectured by Dunfield, Gukov and Rasmussen.

Motivation & Objective

• To establish the triply graded symmetry conjecture of Dunfield, Gukov, and Rasmussen for all knots using a direct algebraic method.
• To define a new family of commuting operators $F_k$ in Khovanov-Rozansky homology analogous to tautological classes in cohomology of character varieties.
• To prove that $F_2$ satisfies the hard Lefshetz property, thereby realizing the conjectured symmetry in homology grading.
• To extend the symmetry to $y$-ified homology for links, showing that $x_i$ and $y_i$ actions are exchanged under the symmetry.
• To provide a conceptual and formal framework using the dg algebra $\mathcal{A}$ and its coproduct structure, avoiding heavy geometric machinery.

Proposed method

• Construct a dg algebra $\mathcal{A}$ with generators $x_i, x'_i, \xi_i, u_k$, equipped with a differential $d$ defined by $d(\xi_i) = x_i - x'_i$ and $d(u_k) = \sum_i h_{k-1}(x_i, x'_i)\xi_i$, where $h_k$ is the complete symmetric function.
• Impose relations $f(x_1,\ldots,x_n) = f(x'_1,\ldots,x'_n)$ for all symmetric functions $f$, making $\mathcal{A}$ an $R$-$R$ bimodule with $R = \mathbb{C}[x_1,\ldots,x_n]$.
• Define a coproduct $\Delta: \mathcal{A} \to \mathcal{A} \otimes_R \mathcal{A}$ that endows tensor products of $\mathcal{A}$-modules with an $\mathcal{A}$-module structure.
• Use the coproduct to define operators $F_k$ on Khovanov-Rozansky homology via the action of $u_k$ on $\mathcal{A}$-modules.
• Prove that $F_2$ satisfies the hard Lefshetz property by analyzing the action of $u_2$ on the homology, leveraging the structure of $\mathcal{A}$ and its homotopy equivalence to $R$.
• Extend the result to $y$-ified homology $\mathrm{HY}(L)$, showing that the symmetry exchanges the actions of $x_i$ and $y_i$ in the module structure.

Experimental results

Research questions

• RQ1 Can the triply graded symmetry in Khovanov-Rozansky homology be proven using a direct algebraic construction rather than geometric representation theory?
• RQ2 Do operators analogous to tautological classes in character variety cohomology exist in Khovanov-Rozansky homology, and do they satisfy hard Lefshetz-type properties?
• RQ3 Is the hard Lefshetz property for $F_2$ sufficient to establish the full symmetry conjectured by Dunfield, Gukov, and Rasmussen?
• RQ4 How can the symmetry be extended from knots to links, particularly in the $y$-ified homology framework?
• RQ5 What is the algebraic structure of the dg algebra $\mathcal{A}$, and how does its coproduct enable the construction of commuting operators on homology?

Key findings

• The paper proves that $F_2$ satisfies the hard Lefshetz property in Khovanov-Rozansky homology, which implies the full symmetry of the triply graded homology as conjectured by Dunfield, Gukov, and Rasmussen.
• The construction of the operators $F_k$ via the dg algebra $\mathcal{A}$ and its coproduct provides a direct, formal algebraic proof of the symmetry, bypassing the need for advanced geometric machinery.
• For knots, the symmetry $\dim\overline{\mathrm{HHH}}_{i,-2j,k} = \dim\overline{\mathrm{HHH}}_{i,2j,k+2j}$ holds universally, as $F_2$ induces an isomorphism between the corresponding graded components.
• In the $y$-ified homology $\mathrm{HY}(L)$, the symmetry exchanges the actions of $x_i$ and $y_i$, generalizing the result to links with multiple components.
• The dg algebra $\mathcal{A}$ is a free resolution of $R$ over $R \otimes_{R^{S_n}} R$, with $H_0(\mathcal{A}) \simeq R$ and higher homologies vanishing, ensuring the algebraic consistency of the construction.
• The framework is motivated by, but does not require, a direct geometric link to character varieties; instead, it formalizes the algebraic analogues of tautological cohomology classes via $\mathcal{A}$.