Virtual-knot problem 36 — electrical

Definitions and mathematical statement

Let a signed graph be a graph $G$ together with edge signs $\varepsilon:E(G)\to\{+1,-1\}$. The source passes from classical knot diagrams to signed planar graphs using checkerboard and medial constructions, then extends the graph moves to abstract, possibly nonplanar graphs. Let $\sim_{\mathrm{el}}$ be the intended electrical equivalence generated by the source's loop/edge replacements (written "pedant loops and edges"), series and parallel replacements, and star--triangle exchanges, with their required sign and conductance rules. Those precise local rules and their admissibility conditions are not supplied. The proposed theory is \[ EG_{\pm}=\{\text{intended signed graphs}\}/\!\sim_{\mathrm{el}}. \]

With two chosen nodes and signs interpreted as generalized electrical conductances, the source states that conductance between the nodes is invariant under knot isotopy restricted to the complement of those nodes. It further states that conductance remains an invariant of the abstract electrical equivalence classes, demonstrating nontriviality. The choice and preservation of nodes, coefficient system, and treatment of singular or undefined conductances must be specified in a precise model.

Problems. Find new invariants of these graph-knots, namely well-defined functions on the intended electrical equivalence classes beyond the already described conductance invariant. Explore their relation with virtual knots: a planar immersion of a signed graph induces cyclic orders of incident edges at its nodes and hence an associated medial-type virtual diagram. Determine the mathematical relationship between this construction and electrical equivalence, including its dependence on immersion and cyclic orders. The source describes the electrical theory as similar in spirit to, but quite different from, virtual knot theory. It does not assert that the immersion construction already descends to a well-defined map $EG_{\pm}$ to virtual-knot classes, nor that the theories have the same equivalence relation. General medial constructions can also produce multiple components, so the knot-versus-link domain must be checked.

Short English statement

Develop new invariants of signed graphs modulo electrical moves and investigate how their planar immersions produce virtual diagrams. The source preserves conductance with chosen nodes, but omits exact move rules and does not establish an immersion-independent correspondence with virtual knots.

Sources

• Local UnsolvedMath registry, permanent ID 10600036, source code catalogue entry, "Virtual-knot problem 36 --- electrical." No individual proposer is supplied.
• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory, revised survey, problem-list item 36. Supplied URL: https://arxiv.org/abs/1409.2823; recorded access 2026-07-29; extraction source-tex. The dedicated section contains 60 top-level enumerated items and retains compound items intact. The opening reference for the graph-knot/conductance construction is missing from the extraction.
• Supplied status open and OPEN-TRIAGE. The research summary mentions the classical electrical-network paradigm, Kirchhoff, conductance, and star--triangle moves, and reports no dedicated published follow-up located for "electric knots." This is a supplied search assessment, not an independently verified absence claim. Current primary-literature status and redistribution terms of the publicly accessible author/publisher copy remain NEEDS_REVIEW.