Proof of a conjecture of Fomichev and Karev
Qi Yan, Qingying Deng, Xian'an Jin
TL;DR
The paper proves the conjecture of Fomichev and Karev by establishing the equality of two graph invariants, φ and ψ, for every simple graph. It rewrites ψ through a combinatorial transformation that counts kernel vectors over the binary field 𝔽₂ and pairs them with vertex-subset indicators, enabling a sum interchange. A parity argument links the condition U ⊆ S(U) to the induced subgraph G|U being Eulerian, yielding a φ-like Eulerian-subgraph expression for ψ. Consequently, ψ(G) = φ(G) for all G, connecting the sl(2) weight system with three-color colorings and confirming the conjecture. This work deepens the bridge between knot-theoretic weight systems and classical graph invariants.
Abstract
We prove a conjecture of Fomichev and Karev [{European J. Combin.} 127 (2025) 104160] by showing the equality of two graph invariants: $\varphi$, defined via graph colorings, and $ψ$, derived from the $\mathfrak{sl}(2)$-weight system of its 2-dimensional irreducible representation.