AlgorithmAlgorithm: AI-adaptedProof: Not applicablePipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Breadth-first Reidemeister search

Definition

Input: Two valid normalized oriented planar-diagram codes $C_{0}, C_{1}$ for connected one-component knot diagrams.

Output: The algorithm returns Found(path) only if it has a finite sequence of code classes from $C_{0}$ to $C_{1}$ , with a finite local-move certificate for each adjacent pair. It has no negative-output branch. It may run forever if the target code class is never reached.

Instructions:

Validate both input codes using the planar-code validation procedure. If either is invalid, return that first invalid-code result. For each valid code, form its canonical representative by checking all bijections of its finite crossing labels and edge labels, recomputing the prescribed edge normalization, and selecting the lexicographically least resulting code.
Set the initial frontier to the singleton containing the canonical representative of $C_{0}$ . Initialize a discovered-code set with that representative and an empty predecessor record.
At the beginning of each breadth layer, test every code in the current frontier against the canonical representative of $C_{1}$ . If one matches, follow its stored predecessor records back to $C_{0}$ , reverse the resulting finite list, and return the path with its move certificates.
Otherwise, for each code $C$ in the current frontier, call the Reidemeister-neighbor algorithm. For each returned pair $(E, μ)$ , canonicalize $E$ by the finite relabeling procedure in step 1. If its canonical representative has not been discovered, add it to the next frontier and store $(C, μ)$ as its predecessor record.
Replace the current frontier by the next frontier and repeat from step 3. If a layer is empty, continue with empty layers; do not return “inequivalent.”

Code equality in this procedure means equality of canonical representatives under the finite crossing/edge relabelings of the normalized code convention. Every loop in one layer is finite; the number of layers has no fixed bound.

Definition references

Planar-diagram, Gauss, and signed Dowker–Thistlethwaite codes

Planar-code validation and diagram reconstruction

Enumerate Reidemeister move neighbors from a knot code

Calegari, Introduction to Knot Theory (2025), §§1.2–1.3

Roberts, Knots Knotes (1999), §2.3

Depends on

Used by

A finite failed move search is not a negative certificate Counterexample
Reidemeister enumeration is only a semidecision Theorem

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Danny Calegari, Introduction to Knot Theory, University of Chicago course notes (2025), §§1.2–1.3 (standard reference, not scraped)
Justin Roberts, Knots Knotes, Edinburgh course notes (1999), §2.3, pp. 10–13 (standard reference, not scraped)