Bar-Natan Homology for Involutive Links

• The theory refines Bar-Natan’s homology by integrating involution symmetry into a Borel-equivariant complex that yields triply graded invariants.
• It defines new numerical invariants via a mixed grading system, offering refined equivariant slice genus bounds for cobordisms.
• The refined invariants distinctly differentiate between equivariant and isotopy-equivariant slicing, providing novel insights into knot concordance and symmetries.

Bar-Natan homology for involutive links refers to the refinement and extension of Bar-Natan’s homological invariants (originally developed as a deformation of Khovanov homology) to the setting of links equipped with a symmetry of order two (an involution). The involutive context is classically motivated by research on periodic and strongly invertible or involutive knots, where additional structure—such as an orientation-reversing or orientation-preserving link symmetry—plays a central role. The recent refinement of Bar-Natan homology for involutive links introduces new algebraic invariants, triply-graded structures, and powerful genus bounds for equivariant cobordisms, placing Bar-Natan theory at the forefront of equivariant and categorified knot theory (Borodzik et al., 18 Jul 2025).

1. Algebraic Framework: Borel-Equivariant Bar-Natan Complex

For a link $L$ in $S^3$ with a fixed involution $\tau$, the standard Bar-Natan chain complex $\mathcal{C}_{\mathrm{BN}}(L)$ of $L$ (over a ground field $\mathbb{F}$, typically $\mathbb{F}_2$ or $\mathbb{Q}$, with variable $u$ of degree $-2$) carries an induced action of $\tau$. The core algebraic step is forming the Borel-equivariant or "Borelified" complex,

$$_Q(L) = \mathcal{C}_{\mathrm{BN}}(L) \otimes_{\mathbb{F}} \mathbb{F}[Q]$$

with differential

$\partial_Q = \partial + Q \cdot (1 + \tau).$

Here $Q$ is a formal variable, typically of degree $-1$. This construction packages the involutive symmetry directly into the chain-level definition in a way that resembles the Borel construction in equivariant cohomology. The resulting chain complex $(Q(L), \partial_Q)$ is triply graded (in the homological, "quantum," and $Q$-degrees), and recovers the original Bar-Natan complex by taking the invariants under a "forgetful" functor ($Q=0$) (Borodzik et al., 18 Jul 2025).

2. Triply Graded Structure and Refined Invariants

The Borel-equivariant Bar-Natan complex gives rise to a triply graded homology theory. The machinery enables the definition of refined slice-genus bounds and other numerical invariants sensitive to the involutive structure. The most fundamental such invariant is

$$\tilde{s}_Q(K, \tau) = \max \left\{ i \mid \exists\,[x] \in H_*(Q(K)) \text{ with } \iota_*[x] = u^{-i/2} \right\},$$

where $\iota_*$ is the localization map after inverting $u$. A further refinement utilizes truncations in the $Q$-variable:

$$\tilde{s}_{Q,A,B}(K, \tau) = \max \left\{ i \mid \exists\,[x] \in H_*(Q(K)/Q^B) \text{ with } \iota_*[x] = u^{-i/2}Q^A \right\},$$

for integers $0 \leq A < B$. These invariants form a suite of genus bounds for equivariant or isotopy-equivariant slice surfaces (Borodzik et al., 18 Jul 2025).

The complex can be further "mixed" with a Lobb–Watson-style (axis or position-based) grading, introducing a new variable $W$ for this additional (usually half-integral) grading, leading to deep trigraded invariants that track both equivariant and geometric data.

3. Equivariant Genus Bounds and the Distinction Between Equivariant and Isotopy-Equivariant Slicing

A principal achievement is the derivation of sharp inequalities on the minimal genus $g$ of an equivariant cobordism between involutive links. For a cobordism $\Sigma$ from $(K_1, \tau_1)$ to $(K_2, \tau_2)$,

$$\tilde{s}_{Q,A,B}(K_1,\tau_1) - 2g(\Sigma) \leq \tilde{s}_{Q,A,B}(K_2,\tau_2).$$

Consequently, the equivariant slice genus $g_{\mathrm{eq}}(K, \tau)$ (the minimal genus of a $\tau$-equivariant surface ending on $K$) satisfies

$$g_{\mathrm{eq}}(K, \tau) \geq \frac{1}{2} \left| \tilde{s}_{Q,A,B}(K, \tau) \right|.$$

A crucial discovery is that these invariants can distinguish the "true" equivariant genus, where a slice surface is strictly fixed by the involution, from the isotopy-equivariant genus, where only an isotopy rel boundary is required. Explicitly constructed examples (such as connected sums of strongly invertible knots) show that the gap $g_{\mathrm{eq}} - g_{\mathrm{iso}}$ can be arbitrarily large for certain involutive knots (Borodzik et al., 18 Jul 2025).

4. Functoriality, Spectral Sequences, and Comparisons

The equivariant Bar-Natan theory exhibits functoriality with respect to equivariant cobordisms and induces spectral sequences relating the equivariant and non-equivariant Bar-Natan or Khovanov homologies. The spectral sequence structure is compatible with localization in $u$ and $Q$, and with the additional $W$-grading, providing a filtered invariant structure. The extra symmetry encoded by $\tau$ introduces corrections in the differentials analogous to the role of involutive operators in involutive Floer theories (such as those found in involutive monopole or Heegaard Floer homology) (Borodzik et al., 18 Jul 2025).

Previous attempts at equivariant or involutive Khovanov-type invariants (mapping cone constructions, Lee/Bar-Natan analogues, equivariant spectral sequences as in the work of Sano, Lobb–Watson, and others) detected only the isotopy-equivariant genus or blended equivariant and non-equivariant information (Borodzik et al., 2017), but the present Borel-complex approach yields invariants sensitive to the strict fixed-point symmetry.

5. Interplay with Skein-Theoretic and Quantum Invariants

The construction aligns with and extends classical ideas of categorification for link polynomials, such as the Kauffman bracket—see the foam-based or skein module homologies for links in thickened surfaces (0810.5566). The extra equivariant grading(s) in the Bar–Natan context enable one to probe symmetries in the categorified skein module, potentially yielding richer information than the polynomial invariants alone.

Furthermore, constructions for links in $\mathbb{RP}^3$ and their relation to twisted orientations and covering spaces (Chen, 2023), as well as category-theoretic and immersed-curve approaches (Kotelskiy et al., 2019, Necheles et al., 2022), provide a broader algebraic and geometric framework in which to situate involutive Bar-Natan homology.

6. Applications, Examples, and Future Directions

The refined invariants obtained from the Borel-equivariant Bar–Natan complex are used to construct explicit lower bounds for the equivariant slice genus, to demonstrate non-equivalence with isotopy-equivariant slice surfaces, and to analyze the impact of symmetry on concordance phenomena. The construction suggests possible extensions to higher-order symmetries and to categorifications of other quantum invariants, as well as adaptations to odd and annular settings.

Comparison with analogous invariants in involutive Heegaard Floer and monopole Floer theory (Lin, 2016) highlights new avenues for exploration, such as the development of more general equivariant triply-graded link homologies and the potential for discovering new obstructions to equivariant slicing or concordance.

Summary Table: Key Features of Bar-Natan Homology for Involutive Links

Aspect	Classical Bar–Natan	Involutive Refinement
Chain complex	$\mathcal{C}_{\mathrm{BN}}(L)$ over $\mathbb{F}[u]$	$$Q(L) = \mathcal{C}_{\mathrm{BN}}(L)\otimes \mathbb{F}[Q]$$ with $\partial_Q$
Involution	Not present	$\tau$ acts on the complex
Extra grading(s)	Homological, quantum	Homological, quantum, $Q$-grading, possible $W$-grading
Refined invariants	$s_{\mathrm{BN}}$	$\tilde{s}_Q$, $\tilde{s}_{Q,A,B}$, axis grading
Slice genus bounds	For classical slice	Separate equivariant and isotopy-equivariant genus bounds
Distinguishes	Unknotting/ribbon	Arbitrarily large invariants for equivariant genus differing from isotopy-equivariant genus

7. Broader Context and Open Problems

Bar–Natan homology for involutive links marks a significant development in equivariant (and categorified) link theory. The introduction of the Borel complex and the refined numerical invariants bridge combinatorial, algebraic, and smooth topological phenomena, revealing new subtleties in the interplay between symmetry and slicing. Open directions include explicit algorithmic computation for more complex links, further generalization to higher-order symmetries, relations to other equivariant and odd/annular Khovanov-type theories, and deeper connections with gauge-theoretic and Floer-type invariants.

The establishment of genuine differences between equivariant and isotopy-equivariant slice genera highlights the richness of the equivariant category and motivates further exploration of link concordance and slicing in the presence of symmetries (Borodzik et al., 18 Jul 2025).