[Paper Review] Stable homology of torus links via categorified Young symmetrizers I: one-row partitions

Matt Hogancamp|arXiv (Cornell University)|May 29, 2015

Homotopy and Cohomology in Algebraic Topology4 citations

TL;DR

This paper proves that the triply graded Khovanov-Rozansky homology of the torus link $T_{n,k}$ stabilizes as $k \to \infty$, computing the stable homology as a ring and confirming a conjecture by Gorsky-Oblomkov-Rasmussen-Shende. It constructs categorified Young symmetrizers via Soergel bimodules and identifies the stable homology as a derived endomorphism ring of a limit complex $P_n$, establishing foundational theory for categorical idempotents.

ABSTRACT

We show that the triply graded Khovanov-Rozansky homology of the torus link $T_{n,k}$ stablizes as $k o \infty$. We explicitly compute the stable homology (as a ring), which proves a conjecture of Gorsky-Oblomkov-Rasmussen-Shende. To accomplish this, we construct complexes $P_n$ of Soergel bimodules which categorify the Young symmetrizers corresponding to one-row partitions and show that $P_n$ is a stable limit of certain Rouquier complexes. A certain derived endomorphism ring of $P_n$ computes the aforementioned stable homology of torus links. Along the way establish some general theory of categorical idempotents.

Motivation & Objective

• To prove the stabilization of triply graded Khovanov-Rozansky homology for torus links $T_{n,k}$ as $k \to \infty$.
• To compute the stable homology as a ring, confirming a conjecture by Gorsky-Oblomkov-Rasmussen-Shende.
• To construct complexes $P_n$ of Soergel bimodules that categorify Young symmetrizers for one-row partitions.
• To establish a general theory of categorical idempotents in the context of Soergel bimodules and Rouquier complexes.

Proposed method

• Construct complexes $P_n$ of Soergel bimodules that categorify Young symmetrizers for one-row partitions.
• Show that $P_n$ arises as the stable limit of certain Rouquier complexes under increasing $k$.
• Identify the stable Khovanov-Rozansky homology of $T_{n,k}$ with the derived endomorphism ring $\operatorname{End}^\bullet_{\mathcal{D}}(P_n)$ in the derived category.
• Use the structure of Soergel bimodules and their categorified idempotents to analyze the limit behavior.
• Leverage the categorification of Young symmetrizers to control the homological structure of the stable link homology.
• Develop general machinery for categorical idempotents in the context of Soergel bimodule categories.

Experimental results

Research questions

• RQ1 Does the triply graded Khovanov-Rozansky homology of $T_{n,k}$ stabilize as $k \to \infty$?
• RQ2 What is the structure of the stable homology ring for torus links $T_{n,k}$ in the limit $k \to \infty$?
• RQ3 Can the Young symmetrizers for one-row partitions be categorified via Soergel bimodules in a way that captures stable link homology?
• RQ4 How do Rouquier complexes relate to the stable limit of link homology for torus links?
• RQ5 What general properties do categorical idempotents in Soergel bimodule categories satisfy?

Key findings

• The triply graded Khovanov-Rozansky homology of $T_{n,k}$ stabilizes as $k \to \infty$, confirming a conjecture by Gorsky-Oblomkov-Rasmussen-Shende.
• The stable homology is explicitly computed as the derived endomorphism ring of the complex $P_n$ of Soergel bimodules.
• The complex $P_n$ categorifies the Young symmetrizer for the one-row partition $(n)$, providing a categorical lift of the idempotent in the group algebra.
• The stable limit of Rouquier complexes for $T_{n,k}$ converges to $P_n$, establishing a link between categorified representation theory and link homology.
• A general theory of categorical idempotents is developed, with $P_n$ serving as a key example of a stable idempotent in the Soergel bimodule category.
• The stable homology ring structure is fully determined by the derived endomorphism ring of $P_n$, providing a complete algebraic description.