bibkey: kuperberg2026oddvas
authors: Greg Kuperberg
year: 2026
title: Odd Vassiliev invariants vanish at four loops
doi: 10.48550/arXiv.2608.11334
url: https://arxiv.org/abs/2608.11334v1
claim: Defines the rational connected Jacobi quotients with commuting colored legs, gives their symmetric-power coefficient description and connected sum, proves odd vanishing for uncolored four-loop diagrams, and explicitly asks whether the marked four-loop population (1, ell-1) always vanishes for odd ell.
strata_touched: []
license: citation-only
triage: anchor

Odd Vassiliev invariants vanish at four loops

Greg Kuperberg, Odd Vassiliev invariants vanish at four loops, arXiv:2608.11334v1. The primary article’s Theorem 1.1 establishes ${\mathcal{P}}_{4,\ell }=0$ for odd $\ell$ in the uncolored AS/IHX quotient. Section 2.1 defines the colored quotient by AS, IHX, and CL. Ordinary legs of different colors commute; numbered anchors do not commute with ordinary legs or with each other. A distinguished leg of a singleton ordinary color is therefore different from an anchor.

Section 3.1, Lemma l:sympow, identifies commuting leg populations with tensor products of symmetric powers of relative first cohomology. The relative subset consists of the true numbered anchors. Equation e:gamma4 gives the common central contraction underlying IHX coefficient transport; e:dual and Section 3.3 describe the dual weight-polynomial viewpoint. Section 3.2, Lemma l:consum, gives connected sum at a singleton ordinary color. Corollary c:odd12 establishes odd vanishing at one and two loops with arbitrary ordinary colors. The three-loop discussion explicitly retains nonzero ${\mathcal{P}}_{3,(\ell ,\ell ,\ell )}$ when $\ell$ is odd.

In Section 7, “Five or more loops,” the source says:

We do not know whether ${\mathcal{P}}_{4,(1,\ell -1)}$ always vanishes.

Here $\ell$ is odd. The same paragraph identifies the remaining bridge skeleton as a tetrahedron with a tadpole and points out that ${\mathcal{P}}_{3,(1,1,1)}$ does not vanish.

The two-color exposition uses these definitions and coefficient strategy inside its proof. Its stable auxiliary domain is $g\ge 1$ or at least two true numbered anchors; its one-anchor genus-zero strut and colored-Y counterexamples explain why the unrestricted auxiliary wording of l:sympow is not used. The two-color argument is not attributed to Kuperberg’s uncolored Theorem 1.1, and it does not use arbitrary-color genus-three odd vanishing or the scalar leg-by-digon and leg-by-triangle lemmas.

Verified locator

• Primary article: https://arxiv.org/abs/2608.11334v1.
• Primary HTML: https://arxiv.org/html/2608.11334v1.
• Primary TeX SHA256: 18da8ad7d658cad15bee133f902728c5d9d249b2f6c87b561798e1cf4fe74ead.
• Definitions: §2.1; coefficient contraction: §2.2, e:gamma4; coefficient presentation: §3.1, l:sympow; socket gluing: §3.2, l:consum; duality: e:dual and §3.3; small genera: §4, c:odd12; explicit marked question: §7, “Five or more loops.”