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Holonomicity from a Heegaard-Floer Perspective

Benjamin Cooper, Robert Deyeso

TL;DR

The paper studies holonomicity of colored knot invariants from a Heegaard-Floer perspective. It defines $S^r$-colored knot Floer homologies $S^r\widehat{CFK}(K)$ via Rozansky-style infinite-braid cabling and proves these sequences are homologically $q$-holonomic by constructing Koszul-type resolutions in a Weyl-algebra framework. It also shows that the $S^r$-colored Alexander polynomials $\Delta^r_K(q)$ are $q$-holonomic in the classical sense, with explicit recurrences built from $D$-operators and the unknot contribution, and that the graded Euler characteristics of the colored knot Floer homologies recover these polynomials. Together, these results provide categorified recurrence relations for colored knot invariants, linking Heegaard–Floer theory to $q$-holonomicity and to AJ-type phenomena in quantum topology, and suggesting robust structural control over color-dependence in knot invariants.

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.

Holonomicity from a Heegaard-Floer Perspective

TL;DR

The paper studies holonomicity of colored knot invariants from a Heegaard-Floer perspective. It defines -colored knot Floer homologies via Rozansky-style infinite-braid cabling and proves these sequences are homologically -holonomic by constructing Koszul-type resolutions in a Weyl-algebra framework. It also shows that the -colored Alexander polynomials are -holonomic in the classical sense, with explicit recurrences built from -operators and the unknot contribution, and that the graded Euler characteristics of the colored knot Floer homologies recover these polynomials. Together, these results provide categorified recurrence relations for colored knot invariants, linking Heegaard–Floer theory to -holonomicity and to AJ-type phenomena in quantum topology, and suggesting robust structural control over color-dependence in knot invariants.

Abstract

We construct -colored knot Floer homologies and prove that they satisfy categorified recurrence relations. The associated Euler characteristic implies -holonomicity of the corresponding sequence of colored Alexander polynomials, in analogy with the AJ conjecture for colored Jones polynomials.
Paper Structure (19 sections, 21 theorems, 110 equations, 10 figures)

This paper contains 19 sections, 21 theorems, 110 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Holonomicity of colored Alexander polynomials
  3. $S^n$-colored Alexander polynomials
  4. Recurrence relation for $S^r$-colored Alexander polynomials
  5. The Weyl algebra
  6. $D$-operators
  7. Incorporating the unknot $U$
  8. Overview of Heegaard-Floer theory
  9. The curve invariant
  10. Knot Floer Homology
  11. Vertical Simplification
  12. Cabling immersed curves
  13. Colored knot Floer homology
  14. Weyl algebra and homological holonomicity
  15. Some algebraic definitions
  16. ...and 4 more sections

Key Result

Theorem 1.2

There exists a notion of homological $q$-holonomicity for sequences of (filtered) chain complexes of graded vector spaces. For each knot $K$ in $S^3$, the $S^r$-colored knot Floer homologies $\{S^r\widehat{\textit{CFK}}(K)\}_{n\geq 1}$ is homologically $q$-holonomic.

Figures (10)

  • Figure 1: The invariants for the unknot $U$ and right-handed $T(2,3)$, lifted to $\overline{T}_M$. Integral heights between the marked points $\bullet$ are indicated underneath.
  • Figure 2: To construct $C^-$$w$-marked points are replaced with $(z,w)$-pairs of marked points
  • Figure 3: Bigons from $x$ to $y$ and from $y$ to $x$, see Ex. \ref{['ex:bigons']}
  • Figure 4: A cabling pattern. A minimal box is $2g(K)$ wide.
  • Figure 5: The $T(3,4)$ cabling applied to the pattern in Fig. \ref{['fig:pattern']}.
  • ...and 5 more figures

Theorems & Definitions (66)

  • Conjecture 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • ...and 56 more