Doubly-graded exponential growth of colored torus knot homology

TL;DR

This paper constructs a new invariant for colored torus knots and links, demonstrating exponential growth in the invariant with respect to color, and verifies related conjectures in knot theory.

Contribution

It introduces a recursive formula for reduced HOMFLY homology of colored positive torus knots and links, and proves exponential growth, resolving a longstanding conjecture.

Findings

Doubly-graded invariant grows exponentially with color

Recursive formula for reduced HOMFLY homology of torus knots

Verification of the color-shifting conjecture in multiple examples

Abstract

We give an invariant construction of reduced HOMFLY homology for arbitrary links reduced at components of arbitrary color and prove some structural properties relating this invariant to unreduced HOMFLY homology. Combined with previous results, this gives a recursive formula for the reduced HOMFLY homology of colored positive torus knots and some colored positive torus links. We prove that after forgetting the quantum grading, the resulting doubly-graded invariant of positive torus knots grows exactly exponentially in the color, resolving a 2013 conjecture of Gorsky--Gukov--Sto\v{s}i\'c. Finally, we verify the ``color-shifting" conjecture of Gukov--Nawata--Saberi--Sto\v{s}i\'c--Su{\l}kowski in many examples.