Holonomicity from a Heegaard-Floer Perspective

TL;DR

This paper introduces $S^r$-colored knot Floer homologies, demonstrating their categorified recurrence relations and establishing the $q$-holonomicity of related colored Alexander polynomials, drawing parallels with the AJ conjecture.

Contribution

It constructs new $S^r$-colored knot Floer homologies and proves their recurrence relations, linking them to $q$-holonomicity of colored Alexander polynomials.

Findings

$S^r$-colored knot Floer homologies satisfy categorified recurrence relations

The Euler characteristic shows $q$-holonomicity of colored Alexander polynomials

Analogy with the AJ conjecture for colored Jones polynomials

Abstract

We construct $S^r$-colored knot Floer homologies and prove that they satisfy categorified recurrence relations. The associated Euler characteristic implies $q$-holonomicity of the corresponding sequence of colored Alexander polynomials, in analogy with the AJ conjecture for colored Jones polynomials.