OEIS A343358 extended: a(14) = 17,831 and a(15) = 74,581 connected realizable circle graphs, with the graphs themselves

OEIS A343358 counts the connected graphs on n vertices that are realizable as circle (chord-interlacement) graphs of Gauss diagrams of closed planar curves with n crossings (Vernitski; Khan–Lisitsa–Lopatkin–Vernitski, arXiv:2108.02873v1); known terms n = 3..13, keyword hard. This archive gives the next two terms, a(14) = 17,831 and a(15) = 74,581, and the complete lists of the graphs for n = 13, 14, 15 in canonical graph6 form, each with the Burton-census names of the prime alternating knots whose minimal diagram realizes it (4,573 / 17,831 / 74,581 data lines). REDUCTION: a connected realizable graph is the interlace graph of a reduced prime alternating diagram, hence of a minimal diagram of a prime alternating knot with n crossings (Kauffman–Murasugi–Thistlethwaite, Menasco); all such diagrams of one knot are flype-related (Menasco–Thistlethwaite), and a flype does not change the labelled interlace graph (stated by Khan et al., Section 9; proof and counter-case test in the archive). So a(n) is the number of interlace-graph classes over one minimal diagram per prime alternating knot. The lemma was counter-case-tested on 62,312 flype-related pairs of distinct minimal diagrams of the same knot (KnotScape vs Burton tables, n = 13, 14 and 15): 0 counter-cases. COMPUTATION: Burton's census DT codes, SageMath canonical labels; KnotScape's independently decoded tables give identical graph-key multisets at n = 13, 14, 15; n = 13 reproduces the published 4,573. Consistency check (not an independent count) by mutation classes, the route of Lisitsa–Vernitski (Examples and Counterexamples 6 (2024) 100162), who checked a(13) = 4,573 = 4,878 − 305 against Stoimenow's lists, where 305 = 574 alternating knots in mutant families − 269 families; the same subtraction on his 14-crossing list also gives 17,831, a consistency check that holds because our graph groups equal his mutant groups set by set. NEW IN VERSION 2: the two groupings should coincide for a reason, Chmutov–Lando's Theorem 2 (arXiv:0704.1313v1): two chord diagrams have the same intersection graph if and only if they are related by a sequence of mutations of shares; the step from share mutations of the Gauss chord diagrams to Conway mutations of the knot diagrams is not taken from them and not proven here, the agreement with Stoimenow's mutant groups is measured. Our graph groups equal Stoimenow's alternating mutant groups at n = 13 and 14 index set by index set; at n = 15 his list has 8,431 groups, all equal to ours, and our 8,432nd group is one pair absent from his list, 15a_13571 / 15a_13629: Conway mutants by an explicit mutation, and distinct by their verified isometry signatures (SnapPy, verified=True). Stoimenow's table page now lists the pair as a correction (dated 10/26/26). His uncorrected list gives 85,263 − 10,681 = 74,582; with the pair, 85,263 − 10,682 = 74,581. Every count is the output of the included scripts run with open tools (SageMath 10.9, SnapPy 3.3.2, Regina 7.4). Work by Werner Alois Stanggassinger in collaboration with Claude Code (Anthropic): the choice of problem, the direction of the work and the computing hardware are due to the author (a sculptor); the scripts in the archive were written and run by Claude Code (Anthropic) working with the author, and the reduction lemma was worked out by Claude Code under the author's direction. The collaboration is stated as a fact, not a disclaimer. Code GPL-3.0-or-later, data and text CC BY 4.0. See README.md and SHA256SUMS in the archive.