Equivariant Unknotting Number and Involutive Khovanov Homology

Abstract: We demonstrate that the equivariant unknotting number $\widetilde{u}(K)$ of a strongly invertible knot $K$ is bounded below by the $H$-torsion order $\widetilde{\mathrm{ord}}(K)$ of the involutive Bar-Natan homology $\widetilde{\mathrm{BN}}(K)$. This result serves as an equivariant analogue to the bound established by Alishahi. As an application, we identify five strongly invertible prime knots with crossing numbers at most $9$ for which the strict inequality $u(K) < \widetilde{u}(K)$ holds.

• The paper introduces a lower bound for the equivariant unknotting number using the maximal H-torsion order from involutive Bar-Natan homology.
• It employs a mapping cone construction and explicit chain homotopies to rigorously track changes during equivariant crossing changes.
• Numerical examples on strongly invertible knots show that the invariant detects symmetry-sensitive obstructions that classical invariants miss.

Equivariant Unknotting Number and Involutive Khovanov Homology: An Expert Summary

Introduction and Context

The paper "Equivariant Unknotting Number and Involutive Khovanov Homology" (2604.08981) addresses the interaction between the equivariant unknotting number of strongly invertible knots and the $H$-torsion order in the involutive Bar-Natan (equivariant Bar-Natan) homology. Situated within the broader developments surrounding knot homology theories—most notably Khovanov and Bar-Natan homology—the work extends a lower bound for the classical unknotting number due to Alishahi to the setting where an involution symmetry is imposed. The introduction provides a precise technical roadmap linking involutive structures with combinatorial and homological knot invariants.

Technical Framework

Strongly Invertible Knots and Equivariant Unknotting

A strongly invertible knot $(K,\tau)$ is a knot $K \subset S^3$ equipped with an orientation-preserving involution $\tau$ such that $\tau(K) = r(K)$, where $r(K)$ is $K$ with reversed orientation. The equivariant unknotting number $u_\mathrm{eq}(K)$ is defined in terms of the minimal number of self-intersections in a $\tau$-equivariant homotopy unknottings, with precise accounting for three distinct types of equivariant crossing changes (Type A, B, C), as formalized in recent works such as [Boyle-Chen:2026].

Bar-Natan Homology and Its Involutive Analog

Bar-Natan's deformation of Khovanov homology produces, for each knot diagram, a chain complex over $F[H]$. The $(K,\tau)$0-torsion in the resulting homology modules encodes subtle geometric information, yielding lower bounds on the unknotting number. The involutive (or equivariant) Bar-Natan homology $(K,\tau)$1, as introduced by Sano [Sano:2025], is a categorified invariant respecting the involution $(K,\tau)$2, constructed via a mapping cone formalism involving the involution-induced chain maps.

The Main Homological Invariant

The key quantity, the maximal $(K,\tau)$3-torsion order $(K,\tau)$4, is defined as the maximal minimal $(K,\tau)$5 such that $(K,\tau)$6 for an $(K,\tau)$7-torsion element $(K,\tau)$8 in the equivariant Bar-Natan homology.

Main Results

Lower Bound Theorem

The central result is that for a strongly invertible knot $(K,\tau)$9,

$K \subset S^3$0

where $K \subset S^3$1 is as above and $K \subset S^3$2 is the equivariant unknotting number. This generalizes Alishahi's bound on the classical unknotting number [Alishahi:2019] to the equivariant context. The proof proceeds by associating to each equivariant crossing change a controlled homological map between the relevant equivariant chain complexes, analyzing the change in maximal $K \subset S^3$3-torsion order via explicit chain homotopies and naturality properties of the involutive formalism (Section 3).

Functorial Construction and Chain Homotopies

The formal construction uses detailed diagrammatic and categorical manipulations:

• The mapping cone $K \subset S^3$4 is developed for every transvergent (symmetry-respecting) diagram, ensuring invariance up to chain homotopy classes under involutive Reidemeister moves.
• For each crossing change of Types A, B, C, bimodule maps and explicit homotopies are constructed such that the algebraic change in order is quantifiable (see Proposition 3.1 and Lemma 3.1).
• These constructions are strictly equivariant in type, with rigorous tracking of homotopies and their coherence with respect to the chain-level involution.

Numerical and Structural Examples

Computations for all prime strongly invertible knots with up to 9 crossings reveal five explicit examples (including $K \subset S^3$5) for which the equivariant lower bound is strictly stronger than the ordinary one: $K \subset S^3$6 while $K \subset S^3$7, leading to $K \subset S^3$8. The data verifies that the refined invariant captures extra difficulty in unknotting when the symmetry is respected, and demonstrates that the equivariant homology—the main novelty—detects phenomena invisible to its classical counterpart. Strong equivariant effects are also visible in knots with multiple involution types, as illustrated by the difference between $K \subset S^3$9 and $\tau$0.

Theoretical and Practical Implications

This work places the equivariant Bar-Natan $\tau$1-torsion order as a robust lower bound for symmetric knot-unknotting processes, providing a new computable obstruction in equivariant knot theory. In particular, it offers:

• Algorithmic leverage: The computation of $\tau$2 can be automated using modified Khovanov homology algorithms, enabling systematic enumeration across knot tables.
• Symmetry-sensitive invariants: The distinction between classical and equivariant unkotting numbers established here cannot be detected by ordinary homological or polynomial invariants.
• Extensions to concordance and slice genera: The connections to equivariant Rasmussen invariants and equivariant slice genus (cf. $\tau$3 and related invariants) suggest interaction with 4-dimensional equivariant knot theory, and raise prospects for further applications in equivariant sliceness obstructions.

Future Directions

The categorical machinery introduced—particularly involving strict homotopy-equivariant maps and mapping cones—opens the possibility for systematizing equivariant refinements of other link homology theories, such as Lee or odd-Khovanov, and for exploring deeper connections with equivariant stable homotopy types. Further, algorithmic invariants developed herein may find applications in computational recognition of nontrivial equivariant concordance classes and classification of strongly invertible knots up to higher symmetry.

Conclusion

By extracting and extending the torsion-theoretic methods of Bar-Natan and Alishahi into the equivariant domain, this paper (2604.08981) provides a tightly controlled, algebraic lower bound for the equivariant unknotting number of strongly invertible knots. The construction not only produces new, computable invariants but also demonstrates the nontriviality and independence (with respect to classical invariants) of the equivariant unknotting number in concrete families. Thus, involutive Bar-Natan theory is elucidated as a key tool in the study of symmetric knot theory, with substantial potential for further algebraic and geometric exploration.