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The Name and the Knot

Written 2026-08-20. Claims re-read against their sources on 2026-09-27: 10 checked, 7 confirmed, 2 wrong, 1 unverifiable. By claims-audit controls 2026-09-27 (audit agent + fix agent).

Claims

• CONFIRMED In 1974 Kenneth Perko showed that two ten-crossing entries of Rolfsen's table are the same knot
  https://en.wikipedia.org/wiki/Perko_pair ; https://www.researchgate.net/publication/247030230_On_the_Classification_of_Knots : Perko, "On the classification of knots", Proc. AMS 45 no. 2 (August 1974); the AMS PDF itself returned 403 to a script
• CONFIRMED Rolfsen's printed table carries the remark that 10_162 is equivalent to 10_161
  research/perko-pair/data/rolfsen-appendix-c.json : hand transcription of book page 415 reads "(K. Perko has pointed out that 10_162 is actually equivalent to 10_161.)"
• CONFIRMED SnapPy's shipped 10_162 is the surviving knot 10_161, and its 10_163 is the others' 10_162
  https://pypi.org/project/snappy/ (3.3.2, run by the fix agent) : Manifold('10_161').is_isometric_to(Manifold('10_162')) is True; Manifold('10_163').identify() gives K10n40, the Knot Atlas 10_162 (10n_40)
• CONFIRMED 40 Alexander, 14 Jones and 5 Alexander-with-Jones coincident pairs among 249 knots
  research/perko-pair/verify.mjs : verifier recomputes these and matches the Knot Atlas coincidence classes 249/249
• WRONG Jones fails on 10_25 and 10_56 "which the Alexander polynomial catches"
  https://katlas.org/index.php?title=10_25&action=raw : same_alexander = 10_56, K11a140; page's own ladder puts the pair in the Alexander class; fixed
• WRONG 10_40 and 10_103 need the signature or the volume to be told apart
  public/strata/the-name-and-the-knot/data.json : both have signature 2 and sit together in the "Alexander polynomial with signature" class; only volume (12.889 vs 13.875) separates them; fixed
• CONFIRMED 5_1 and 10_132 share the Alexander and Jones polynomials
  https://katlas.org/index.php?title=10_132&action=raw : same_alexander = 5_1, same_jones = 5_1
• CONFIRMED 9_42 and 10_132 have the same hyperbolic volume, 4.05686022...
  https://pypi.org/project/snappy/ (3.3.2, run by the fix agent) : volumes 4.0568602242 and 4.05686022424; not isometric (is_isometric_to False)
• UNVERIFIABLE 9_42 and 10_132 are commensurable, sharing a covering manifold
  SnapPy 3.3.2 covers up to degree 6 (no common cover) and invariant trace fields via PARI algdep/polredabs (both x^5 - x^4 - 2x^2 + 2x - 1) : same trace field is necessary, not sufficient; no source found; aside removed
• CONFIRMED the isometry signature separates every pair it is defined on, and the volume leaves only 9_42/10_132
  research/perko-pair/verify.mjs : verifier checks 1 and 0 pairs, 24/24 pass

What was done

• [fixed] 10_25/10_56 were said to be caught by the Alexander polynomial and 10_40/10_103 to need the signature or volume; reworded to say Alexander, Jones and signature all fail on both pairs and only the volume (or isometry signature) separates them; dated correction added to the page's apparatus
• [fixed] unsourced "they are commensurable, sharing a covering manifold" removed (not provable here; same invariant trace field only); dated correction added