Quantum Alexander Grading in Knot Invariants

• Quantum Alexander grading is a two-variable refinement that extends the classical Alexander grading by incorporating additional basepoints and intersection data.
• The methodology utilizes quantum deformations of Heegaard diagrams to interpolate between the Alexander and Jones polynomials through specific parameter specializations.
• This framework enables unified categorification approaches, offering deeper geometric insights and practical tools for studying quantum and classical knot invariants.

The quantum Alexander grading is a two-variable refinement of the classical Alexander grading in knot Floer homology, arising in the context of quantum deformations of Heegaard diagrams and topological models for quantum invariants. It encodes additional winding and intersection data associated with auxiliary basepoints and diagonal hypersurfaces in symmetric powers of decorated surfaces, enabling a unified treatment of the Alexander and Jones polynomials and their categorifications.

1. Formal Definition of the Quantum Alexander Grading

Given a knot $K$ presented as the closure %%%%1%%%% of an $n$-strand braid $\beta_n$, the construction begins with a genus $g = n-1$ Heegaard diagram

$$\mathcal{H}_{\beta_n} = (\Sigma, \alpha_1, \ldots, \alpha_{n-1};\ \beta_1, \ldots, \beta_{n-1};\ w, z)$$

augmented with $n$ additional marked points, referred to as $q$–basepoints, yielding a quantum Heegaard surface $\Sigma^q = (\Sigma, q)$ and decorated diagram $$\mathcal{H}^q_{\beta_n} = (\Sigma^q, \alpha, \beta, w, z)$$ (Anghel et al., 13 Jan 2026).

Inside $\operatorname{Sym}^{n-1}(\Sigma)$ the relevant hypersurfaces are:

• $V_z$, $V_w$ — as in classical knot Floer theory, associated to the points $z$ and $w$;
• $V_q = \bigcup_{i=1}^n V_{q_i}$ — hypersurfaces for each $q$-basepoint $q_i$;
• $V_d$ — the diagonal set where two of the $n-1$ coordinates coincide.

A topological Whitney disk $u$ contributes intersection numbers $n_z(u)$, $n_w(u)$, $n_q(u)$ ($u \cdot V_q$), and $n_d(u)$ ($u \cdot V_d$).

The quantum Alexander gradings, as formalized in Definition 6.10 of (Anghel et al., 13 Jan 2026), are specified as follows. Fix a "canonical" generator $x_0 \in \mathbb{T}_\alpha \cap \mathbb{T}_\beta$ such that $A^{HF}(x_0)=0$ and $A^{qHF}(x_0)=n-1$. For any intersection point $x$, select a domain $D$ from $x_0$ to $x$ and set: $A^{HF}(x) = n_z(D) - n_w(D)$

$A^{qHF}(x) = (n-1) + n_q(D) + n_d(D)$

The pair $(A^{HF}, A^{qHF}) \in \mathbb{Z} \times \mathbb{Z}$ is the quantum Alexander grading of $x$.

2. Relationship to the Classical Alexander Grading

The quantum Alexander grading refines and extends the classical Alexander grading of knot Floer theory, which is given by

$A^{HF}(x) = n_z(D) - n_w(D)$

where $D$ is a domain between intersections $x_0$ and $x$. In the quantum version, additional intersection counts, $n_q$ and $n_d$, record winding around the newly introduced $q$-basepoints and crossings of the diagonal hypersurface. The second grading,

$A^{qHF}(x) = n_q(D) + n_d(D) + (n-1)$

therefore encodes extra geometric data controlled by the quantum deformation.

Specializing the quantum parameter $d$ to $d=1$ eliminates this refinement and recovers the ordinary Alexander grading. This demonstrates that the quantum Alexander grading is a true enhancement rather than an unrelated invariant (Anghel et al., 13 Jan 2026, Anghel, 2019).

3. Quantum Deformation Parameter and Grading Variables

Within the construction, two deformation parameters $x$ and $d$ are introduced, corresponding to loops around $p$-punctures and $q$-punctures, respectively, in the topological model of the punctured disc. The local system $\varphi$ sends loops to $x^{\pm1}$ and $d^{\pm1}$.

• The exponent of $x$ in the Lagrangian intersection formula reflects the ordinary Alexander grading $A^{HF}$.
• The exponent of $d$ is the quantum Alexander grading $A^{qHF}$.

This bigrading appears as the exponent structure in graded intersection sums and allows distinct specializations to the Alexander and Jones polynomials using different assignments to $d$ and $x$. Specifically:

• Alexander polynomial: $d=1$ specialization,
• Jones polynomial: $d=-x^{-1}$ specialization.

This parameterization thus realizes Jones and Alexander polynomials as two specializations within the same framework (Anghel et al., 13 Jan 2026, Anghel, 2020, Anghel, 2019).

4. Topological and Quantum Models: Coverings and Graded Intersections

In topological models for quantum and colored Alexander invariants, the quantum Alexander grading is realized algebraically and geometrically through covering spaces of configuration spaces of points in punctured surfaces.

Two-variable covers $\mathbb{Z} \oplus \mathbb{Z}_N$ (or $\mathbb{Z} \oplus \mathbb{Z}$ for the non-colored case) induce gradings on homology,

$$H_*^\rho = \bigoplus_{(i,j) \in \mathbb{Z}^2} H_{*, i, j}$$

with deck transformations by $x$ and $d$ acting as grading shifts. The intersection form

$$\langle -, - \rangle: H_{n,m} \otimes H_{n,m} \to \mathbb{Z}[x^{\pm1}, d^{\pm1}]$$

is sesquilinear and respects this bigrading. Quantum invariants, specifically the colored Alexander polynomials, are then extracted as the result of specializations of this intersection form via assignments $x \mapsto s^2$, $d \mapsto -q^{-2}$ (Anghel, 2019, Anghel, 2020).

5. Unification Theorems and Explicit Examples

A principal result (Theorem 6.14 of (Anghel et al., 13 Jan 2026)) demonstrates that the Lagrangian intersection sum,

$$\Omega^q(\beta_n)(x, d) = (d^2x)^{\frac{w(\beta_n)+n-1}{2}} d^{-(n-1)} \sum_{x \in \mathbb{T}_\alpha \cap \mathbb{T}_\beta} \epsilon_x x^{A^{HF}(x)} d^{A^{qHF}(x)},$$

where $w(\beta_n)$ denotes the braid writhing and $\epsilon_x$ is the signing, satisfies: $$\Omega^q(\beta_n)\big|_{d=1} = \Delta_K(x), \qquad \Omega^q(\beta_n)\big|_{x = -d^{-1}} = J_K(x)$$ Thus the construction unifies the Alexander and Jones polynomials as specializations of a quantum bigraded intersection.

As an explicit example, for the right-handed trefoil (the closure of $\sigma^3 \in B_2$), one has five intersection points and the computation yields (Anghel et al., 13 Jan 2026): $$\Omega^q(\sigma^3)(x, d) = x^2 d^3 \left(-x^{-3} + x^{-2} - x^{-1} + 1 - d\right)$$

• Specializing $d=1$ gives the Alexander polynomial $\Delta_{\text{Trefoil}}(x) = x - 1 + x^{-1}$.
• Specializing $d=-x^{-1}$ yields the Jones polynomial $J_{\text{Trefoil}}(x) = -x^{-4} + x^{-1} + x^{-3}$.

6. Quantum Alexander Grading in Categorification Contexts

In categorification frameworks (e.g., foam-based approaches of Robert–Wagner (Robert et al., 2019)), the quantum Alexander grading structure persists but is encoded within bigraded or trigraded complexes. In the case of $\mathfrak{gl}_0$ link homology, the $q$–grading alone is present and carries the Alexander polynomial variable. The contribution of the quantum Alexander grading is realized via the assignment of quantum degrees to foams and the overall $q^{n_- - |s|}$ shift per cube vertex, while the homological grading supplies the second index. This construction results in a bigraded homology whose graded Euler characteristic is the Alexander polynomial, verifying the correct quantum grading behavior.

7. Significance, Applications, and Broader Connections

The quantum Alexander grading provides a systematic way to interpolate between classical and quantum invariants of knots, clarifying the geometric underpinning of knot polynomials in terms of graded intersections in symmetric powers of surfaces and coverings of configuration spaces. It enables:

• Refinement of knot Floer homology and explicit comparison with quantum invariants such as the Jones and colored Alexander polynomials,
• Construction of unifying two-variable invariants and categorifications,
• A framework for topological and algebro-geometric realization of quantum link invariants via intersection models (Anghel et al., 13 Jan 2026, Anghel, 2020, Anghel, 2019).

A plausible implication is that the tools developed—such as quantum Heegaard diagrams and the quantum Alexander grading—facilitate a deeper understanding of categorification phenomena and may be adaptable to other quantum invariants and representation-theoretic contexts.