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The sl(2)\mathfrak{sl}(2) weight system at c=−3/32c=-3/32 extends to a graph 4-invariant

A mix of human-written and AI-generated textHuman understanding: some partsmath.CO — Combinatoricsmath.GT — Geometric Topologymath.RT — Representation Theory

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya ↗

Version 2 / Oct 17, 2026 / CC BY 4.0

Abstract

A weight system is a function on chord diagrams that satisfies the four-term relations. The Lie algebra sl(2)\mathfrak{sl}(2) gives a weight system with values in the polynomial ring Q[c]\mathbb{Q}[c], and its value on a chord diagram depends only on the intersection graph of the diagram. Fomichev, Karev, Pavutnitskiy and Usanov showed that this weight system does not come from a function on graphs, and that a specialisation at a value c=c0c=c_0 can come from one only for c0=0c_0=0, 3/83/8, 11 or −3/32-3/32. The first three values were settled before. They conjectured that the last one, c=−3/32c=-3/32, which is the eigenvalue of the Casimir element on the oscillator representation, also extends. We prove this. The extension is the function ξ(G)=(−32)−nSG(−2)\xi(G)=(-32)^{-n}S_G(-2), where GG is a graph with nn vertices and SG(N)S_G(N) is a signed count of the maps that send each vertex of GG to a two-element subset of {1,…,N}\{1,\dots,N\}: a map counts with the sign −1-1 for every edge whose two endpoints are sent to subsets meeting in exactly one element. The count is a polynomial in NN, so its value at N=−2N=-2 makes sense. The four-term relation for ξ\xi is proved by an involution on these maps. The identity with the weight system is proved by letting the chord diagram act on Verma modules, then on the oscillator representation, where a supertrace on the Weyl algebra turns the value into a sum over perfect matchings, and finally by reading the same sum as a trace in a Clifford algebra on NN generators.

Provenance statement

The proofs, the text, the figures, the Lean 4 formalization and the Python, Julia, C and C++ programs were carried out by Deep Bhattacharjee with the help of AI assistance (Claude, Anthropic) in the Claude Code environment. The paper proves the conjecture of D. Fomichev, M. Karev, F. Pavutnitskiy and S. Usanov (arXiv:2607.24217, Section 4) that the sl(2) weight system specialised at c = -3/32 extends to a 4-invariant of graphs, with the explicit extension xi(G) = (-32)^{-n} S_G(-2): the chord element acts on Verma modules, the Verma modules of highest weights -1/2 and -3/2 are realised in the oscillator representation, the resulting scalar is a supertrace on the Weyl algebra computed by a Wick formula, the sum over matchings is a trace in a Clifford algebra, and the four-term relation follows from an involution on the assignments. Both parts of the conjecture, the four-term relation for xi and the identity w_D(-3/32) = xi(Gamma(D)) for every chord diagram D, are proved in Lean 4 with Mathlib (SL2W.main_theorem, 16 files, 4120 lines), depending only on the axioms propext, Classical.choice and Quot.sound, with no sorry, native_decide or added axiom; programs in Python, Julia, C and C++ check the theorem on all chord diagrams with at most six chords and reproduce the values tabulated by Fomichev et al. The binomial expansion of S_G(N) (part (a) of the main theorem) is proved by hand and not formalized. No statement rests on a floating-point computation. The earlier value c = 3/8 is due to Fomichev and Karev and to Yan, Deng and Jin and is credited to them. Both authors approved the final version. Tools used: Anthropic - Claude (Claude Code); Lean community - Lean 4 (v4.34.1) with Mathlib; Python; Julia; C; C++; shell. arXiv identifier: none (not posted on arXiv). The earlier versions and their checking programs are archived on Zenodo under the concept DOI \href{https://doi.org/10.5281/zenodo.23312529}{10.5281/zenodo.23312529} (versions \href{https://doi.org/10.5281/zenodo.23312664}{10.5281/zenodo.23312664}, \href{https://doi.org/10.5281/zenodo.23384025}{10.5281/zenodo.23384025}, \href{https://doi.org/10.5281/zenodo.23541142}{10.5281/zenodo.23541142} and \href{https://doi.org/10.5281/zenodo.23630528}{10.5281/zenodo.23630528}). The source of this revision is in the repository \url{https://github.com/creelie/Paper_7}, release \texttt{v1.0.1}, archived on Zenodo as \href{https://doi.org/10.5281/zenodo.23658136}{10.5281/zenodo.23658136} (concept DOI \href{https://doi.org/10.5281/zenodo.23658071}{10.5281/zenodo.23658071}).

Tools used

Anthropic
Claude CodeVersion Fable 5.1 Medium

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Version history

  1. v1 · 18 pagesSubmitted by Deep BhattacharjeeInitial depositSupersededOct 17, 2026
  2. v2 · 13 pagesSubmitted by Deep BhattacharjeeSubmitted as a new versionCurrentOct 26, 2026