Take the Khovanov complex of a knot and forget everything that does not survive a certain kind of map. What is left is a group, and in October 2026 the paper found torsion in it, and integer-valued invariants that the $s$-invariants cannot see. This page reads that paper, works one example to the end, and says which parts were checked here and which were not.
A knot is slice if it bounds a smooth disc in the four-ball, and two knots are concordant if they differ by a slice knot. Concordance classes form an abelian group, and obstructing sliceness is one of the headline uses of Khovanov homology: Rasmussen’s $s$-invariant, built from it, gave another proof of the Milnor conjecture. Since then many more concordance invariants have been built from Khovanov homology and its relatives.
The paper studies two local equivalence groups built from the Khovanov complex, one by Lewark and one by Dunfield, Schütz and Lipshitz. A group like this is a deliberately coarse summary: it throws away everything about the complex except what survives maps of one specific kind. Rasmussen’s $s$ over any field is a function on these groups. The question the paper answers is how much more structure the groups have than the $s$-invariants see.
Let $\mathbb{Z}[h]$ be polynomials in one variable $h$ of grading $-2$. A knot-like complex over a graded $\mathbb{Z}[h]$-algebra $R$ is a finitely generated cochain complex of free graded $R$-modules which, after inverting $h$, is chain homotopy equivalent to $h^{-1}R$ sitting in homological degree 0. The reduced Bar-Natan complex of a knot is one.
A chain map $\beta: D \to D'$ is local if $h^{-1}\beta$ is a chain homotopy equivalence. Write $D \le D'$ if a local map exists; $D$ and $D'$ are locally equivalent if both $D \le D'$ and $D' \le D$. The local equivalence classes under tensor product form an abelian group $G(R)$, with the dual complex as the inverse. The default is $G = G(\mathbb{Z}[h])$.
The knot-to-complex assignment is a homomorphism from the smooth concordance group to $G$, so anything one learns about $G$ is a statement about what a concordance invariant of this type could possibly be.
A second group, $\widetilde{C}_{LEO}$, keeps three pieces of data at once: a complex $C$ over $\mathbb{Z}$ (the odd Khovanov complex), a knot-like complex $D$ over $\mathbb{Z}[h]$ (the Bar-Natan complex), and an identification of their reductions. $G$ is a direct summand of it.
The paper’s first result is that $G$ has torsion, and the first torsion element is a five-generator complex $D_{2,1,1}$ over $\mathbb{Z}[h]$ (the author of the paper reports that Claude called it a bowtie complex). Its generators and gradings, written (homological, quantum), are:
| generator | $x$ | $y_+$ | $y_0$ | $y_-$ | $z$ |
|---|---|---|---|---|---|
| bigrading | (−1, −2) | (0, 2) | (0, 0) | (0, −2) | (1, 2) |
Two checks you can do on paper. First, $d^2 = 0$: $d^2(x) = 4h(-4hz) + h^2(8z) + 8(h^2 z) = (-16 + 8 + 8)h^2 z = 0$, and $d^2$ vanishes on the other generators because each has at most one step to $z$. Second, $d$ preserves quantum grading when $h$ is counted as degree $-2$: $d(x)$ has terms of grading $-2$ in every slot.
Knot-like. Invert $h$. The pair $y_-, z$ cancels through $d(y_-) = h^2 z$, and the pair $x, y_+$ cancels through the coefficient $h^2$ in $d(x)$. What remains is $\mathbb{Z}[h,h^{-1}]\langle y_0\rangle$.
Nontrivial. The paper uses $s_c(D) = \max\{j : \exists$ a cycle in bidegree $(0,j)$ that generates $H(h^{-1}D)\}$, a local-equivalence invariant, and the fact that $D$ is trivial exactly when $s_c(D) = s_c(D^*) = 0$. In bidegree $(0,0)$ the cycles are the multiples $k(hy_+ + 2y_0)$, which represent $2k$ times a generator, never a generator. So $s_c(D_{2,1,1}) \ne 0$; in fact it is $-2$, witnessed by the cycle $hy_0 + 4y_-$.
Order 2. Swapping $x \leftrightarrow z$, $y_+ \leftrightarrow y_-$, $y_0 \mapsto -y_0$ identifies $D_{2,1,1}$ with its dual. The dual is the inverse in $G$, so $D_{2,1,1} = -D_{2,1,1}$.
The paper then generalises to a three-parameter family $D_{p,m,n}$ for a prime $p$ and positive integers $m,n$, with $2p^m h^n$ and $2p^{2m}$ in place of $4h$ and $8$. Its Theorem 1.1 says that for each prime $p$ these generate a copy of $(\mathbb{Z}/2)^\infty$ in $G$, independent for different primes. The proof the paper gives for the independence over one prime is an invariant that measures the largest power of $p$ dividing the ideal of values of homology classes; I did not re-derive it.
| Result | Statement, as the paper gives it | Status here |
|---|---|---|
| Thm 1.1 | $G$ has 2-torsion: for each prime $p$ a $(\mathbb{Z}/2)^\infty$ subgroup, independent across primes. | Example $D_{2,1,1}$ re-derived. General case not re-derived. |
| Thm 1.2 | A $(\mathbb{Z}/2)^\infty$ subgroup of the kernel of $\widetilde{C}_{LEO} \to G$. | Not read. |
| Thm 1.3 | For each prime $p$, $G$ has a $\mathbb{Z}^\infty$ direct summand in the kernel of all $s$-invariants. | Not read. The proof imitates Dai–Hom–Stoffregen–Truong over $\mathbb{Z}_p[h]/(ph)$. |
| Thm 1.4 / Cor 3.1 | At least six concordance homomorphisms to $\mathbb{Z}$, independent of the $s$-invariants, $\tau$, and the DHST homomorphisms. | Corollary 3.1 checked on the paper’s own table, §5. |
The proof of Corollary 3.1 in the paper is one sentence: view Table 2 as a matrix and row-reduce it. Table 2 gives fourteen invariants on eleven knots (its caption says thirteen; the header lists fourteen). The rows below are copied from the paper’s table by a script from the extracted text.
| Knot | sQ | sF2 | τ | ψ1o | φ1o | φ2o | φ3o | φ4o | φ5o | φ1e | ψ1e | φ1HF | φ2HF | φ3HF |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31 | 2 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 819 | 6 | 6 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 942 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12n200 | -2 | -2 | -1 | 3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 12n475 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13n309 | 0 | 0 | 0 | -1 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13n4587 | 10 | 10 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 1 |
| 14n19265 | 0 | -2 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | -1 | 0 | 0 | 0 |
| 14n26039 | 0 | 0 | -1 | 1 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 15n5319 | -2 | -2 | -1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | -1 | 0 | 0 |
| 16n956730 | -2 | -2 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | -1 | 0 | 0 |
ch-local-equivalence-verify.py turns the sentence into numbers. Over $\mathbb{Q}$ the full
$11 \times 14$ matrix has rank 11, the eight columns other than the six new invariants
($\psi_1^o,\ \phi_1^o,\dots,\phi_5^o$) have rank 5, so the six add 6 independent directions on these
eleven knots. Since a linear relation among the invariants on all knots would hold on these eleven, that
certifies the six are independent of each other and of the rest. Two controls were run first: zeroing the
$\phi_5^o$ column drops the gain to 5, and replacing it by a copy of $\tau$ also drops it to 5, so the test
can fail. The script also confirms the three relations the paper states for its census: $\tau =
\phi_1^{HF} + 2\phi_2^{HF} + 3\phi_3^{HF}$, $s_{F_2} - s_Q = 2\psi_1^e$, and $\phi_1^e = -\psi_1^e$.
The same script checks the algebra of §3: $d^2 = 0$ and bidegree homogeneity for 64 choices of $(p, m, n)$, with a control that a perturbed differential fails; that $h^{-1}D_{2,1,1}$ has rational homology of rank one in degree 0; $s_c(D_{2,1,1}) = -2$ with the witness a cycle; and self-duality. It also checks that each row of the paper’s Table 3 sums to its printed total, and prints the column sums, which are the numbers of knots of each crossing class that table implies: 249, 185, 888, 5110, 27436, 168030 and 1008906. The script has no knot census to compare them against.
The paper ends with six questions, which I give in my words. Does any knot realise one of the 2-torsion complexes? Can one write explicit homomorphisms from $G$ onto $(\mathbb{Z}/2)^\infty$? Is there torsion of order other than 2? Do the groups have infinitely divisible elements? What is the structure of $G(\mathbb{Z}_p[h]/(ph))$, and is it $\mathbb{Z}^\infty$ with an explicit basis? And is every local equivalence class realised by a knot? The author says the last needs significant new ideas.
A computational remark from the same paper bears on the first question: for every amphicheiral knot up to 16 crossings, and about 40,000 random strongly negatively amphicheiral knots with 19 to 26 crossings, the invariant $s_c$ vanished, so none of the 2-torsion complexes of §3 turned up in that sample.
It makes no claim that local equivalence groups connect to anything else in this series. The paper is about knots and chain complexes; nothing in it is about the operators C, K, F, U or the contact-geometry chapters, and a shared word such as “complex” or “group” is not a shared mechanism. It repeats the author’s statement that the knot concordance results are his own and the AI-assisted research his report; the human side of that is the next chapter.