Enumerate diagrams up to a crossing bound

Input: A nonnegative integer crossing bound $N$.

Output: A finite ordered list of reconstructed knot-diagram candidates. A candidate is a Valid output of the planar-code validation procedure whose code is a normalized planar-diagram code with one oriented component and at most $N$ crossings. Repeated candidates up to crossing relabeling or planar isotopy are allowed; the list is not a list of knot types.

Instructions:

1. If $N$ is not a nonnegative integer, return invalid-bound.
2. Initialize the output list empty. Pass the two normalized crossingless one-component codes, with ccw and cw orientation markers in that order, through the validator and append both outputs first.
3. For each $c=1,\dots ,N$ in increasing order, use crossing labels $\{1,\dots ,c\}$ and the four normalized dart positions at each crossing. Enumerate every crossing-sign vector in lexicographic order, with $-1<+1$. The sign determines the incoming and outgoing positions at each crossing by the normalized-code convention.
4. For each sign vector, enumerate every bijection from the set of its $2c$ outgoing darts to its $2c$ incoming darts in lexicographic order. Pair each outgoing dart with its image to obtain the edge involution, and assign edge labels by the normalized-code rule.
5. Compute the cycles of $\varphi =\alpha \circ \sigma$ for each pairing. For each cycle, in increasing order of its least dart, select that cycle in turn as the outer face and form the corresponding normalized PD code.
6. Pass each code through the planar-code validation procedure. Append its reconstructed output if and only if the validator returns Valid and the directed component tracing has exactly one cycle. Return the list after the finite loops end.

This procedure enumerates normalized finite codes and reconstructed diagrams; it neither identifies equivalent codes nor decides whether two candidates represent isotopic knots.

Planar-diagram, Gauss, and signed Dowker–Thistlethwaite codes — normalized PD codes, local sign arrows, edge labels, and face selectors.

Planar-code validation and diagram reconstruction — validation and reconstruction of each candidate code.

Hass–Lagarias–Pippenger, §§1, 7 — finite diagram inputs and reconstruction context.

Erickson, “Planar Graphs” — rotation systems and face selection.