Exponential growth of colored HOMFLY-PT homology

TL;DR

This paper introduces reduced colored HOMFLY-PT homologies, demonstrating their exponential growth and establishing their relation to sl(N) homologies as N becomes large, confirming key conjectures in link homology theory.

Contribution

It defines reduced colored HOMFLY-PT homologies and proves their large N limit relation to sl(N) homologies, confirming conjectured growth behaviors.

Findings

Verified exponential growth of colored HOMFLY-PT homologies.

Established large N limit correspondence with sl(N) homologies.

Confirmed conjecture of Gorsky, Gukov, and Stošić.

Abstract

We define reduced colored sl(N) link homologies and use deformation spectral sequences to characterize their dependence on color and rank. We then define reduced colored HOMFLY-PT homologies and prove that they arise as large N limits of sl(N) homologies. Together, these results allow proofs of many aspects of the physically conjectured structure of the family of type A link homologies. In particular, we verify a conjecture of Gorsky, Gukov and Sto\v{s}i\'c about the growth of colored HOMFLY-PT homologies.