KeeTaek Kim
6 min
This paper by KeeTaek Kim bridges knot theory and algebraic topology by introducing equivariant unknotting numbers for strongly invertible knots and linking them to involutive Bar-Natan homology. It provides the first lower bounds on these equivariant invariants using torsion in a deformed Khovanov homology theory, with concrete examples showing strict inequalities over classical unknotting numbers.
A strongly invertible knot is a knot K in S³ equipped with an orientation-preserving involution τ such that τ(K) equals the reverse of K (r(K)). The fixed-point set of τ is unknotted by the proven Smith Conjecture: nontrivial finite-order orientation-preserving diffeomorphisms of S³ have unknotted circle fixed sets. These knots admit symmetric unknotting processes respecting τ, unlike general knots.
Intuitively, τ acts like a 'strong symmetry' that reverses orientation while preserving the knot setwise, enabling equivariant study of knot properties. This structure appears in many low-crossing prime knots and is key for applications like detecting slice knots or studying concordance.
The equivariant unknotting number eu(K) is the minimum number of self-intersections in a τ-invariant unknotting isotopy turning K into the unknot. Unlike the classical unknotting number u(K), which counts arbitrary crossing changes, eu(K) requires changes to respect τ-symmetry.
The paper finds examples among prime knots with ≤9 crossings where u(K) < eu(K), meaning symmetry forces more crossing changes. This strict inequality highlights how involution constraints make unknotting 'harder,' with implications for concordance and 4D topology.
Building on Sano's work, involutive Bar-Natan homology ḡBN(K) deforms classical Bar-Natan homology (over R[H]) to incorporate τ-action via a dotted cobordism category Cob^•. For an involutive link (K, τ), it produces a formal complex JKK with involution I_τ.
This homology detects τ-equivariant features missed by Khovanov/Bar-Natan theories. Over F=Z/2Z, it categorifies Jones-like invariants while capturing symmetry via H-torsion, analogous to how Bar-Natan torsion bounds classical unknotting.
Theorem: For strongly invertible K, the maximal ℍ-torsion order ḡord(K) ≤ eu(K). Here, ḡord(K) is the highest power of H dividing torsion in ḡBN(K), bounded above by the Euler characteristic eu(K) (confusingly same notation as unknotting).
This mirrors Alishahi's result ord(K) ≤ u(K) from Bar-Natan torsion. Proof uses τ-invariant cobordisms: each self-intersection in unknotting induces H-multiplication in the complex, so eu(K) self-intersections kill at most H^{eu(K)} torsion, hence the bound.
Applied to all strongly invertible prime knots with ≤9 crossings, it identifies five where u(K) < eu(K), recoverable via [BC26]. Computations use Sano's software, emphasizing practical homology for low-crossing knots.
This work extends torsion-unknotting bounds to equivariant settings, opening doors to symmetry-aware concordance obstructions and higher-rank Khovanov theories.
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.
Alex: So all possible smoothed versions, connected in a grid.
Sam: Exactly. Neighbors—those differing by one choice—link via saddle-shaped surfaces that merge or split circles, like pants connecting time slices. The map between them is the differential; applying it twice cancels out. This setup is the formal Bar-Natan complex. It's invariant under diagram changes, so it truly captures the knot.
Alex: Okay, so persistence through maps between these complexes after crossing changes gives the bound.
Sam: That's the mechanism. For symmetry-respecting changes, the maps are homotopic to H to some power times the identity. Starting from the unknot's zero torsion, each step increases it by at most that power, bounding the steps needed.
Alex: How do they build the symmetry-aware version of that complex?
Sam: They start with the knot diagram and its symmetry rotation, which pairs crossings. For each circle diagram, they connect matching circles under the rotation using cylinders—like tubes linking identical parts in mirrored pictures. This creates an involution map on the complex—applying it twice gets back to the start.
Alex: So it respects the symmetry. Then what?
Sam: They take two copies of the complex and glue them with a map that includes both the direct path and the symmetry flip—like bridging two identical buildings. This forms the involutive Bar-Natan complex, whose homology tracks the symmetric tangles. For a symmetry-safe crossing change, they build special maps between these complexes that multiply torsion by H to a fixed power—two for off-axis changes, one for on-axis.
Alex: So each change dilutes the torsion predictably, like paint fading through safe pipe bends.
Sam: Yes. An algebraic lemma says if complexes link by such maps, their maximal H-torsion orders differ by at most that power. Iterating from the unknot bounds the equivariant unknotting number. For those five knots—like 7_7b—ordinary unknotting is one, but this torsion measure is two, proving symmetry demands more.
Alex: And the same knot shape can have different bounds depending on the symmetry choice, like 7_7a versus b.
Sam: Yes—for 7_7a it's one, but 7_7b and four others hit two. Computations on all prime strongly invertible knots up to nine crossings confirm it, using software on special diagrams. The bounds take the maximum of ordinary unknotting, half the absolute Rasmussen values, and that torsion order.
Alex: What are the limits to these computations?
Sam: They work over mod-two arithmetic. It focuses on knots up to nine crossings and this 180-degree symmetry, not broader ones. Higher mods or bigger knots remain open.
Alex: Fair enough. So this sets up ways to calculate symmetry-respecting untie counts.
Sam: Precisely—a meaningful step for tracking how symmetries complicate knot measures. The constructions and tables make it computable where symmetry matters.
Alex: Well put, Sam. This shows symmetry can force more work to untie a knot. Thanks for breaking it down. Thanks for listening to ResearchPod.