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cat _posts/2026-10-07-lspace-det-sigma-counting-families-by-knot-and-a-counterexample-that-must-be-hyperbolic-of-genus-at-least-3.md

lspace-det-sigma: counting families by knot, and a counterexample that must be hyperbolic of genus at least 3

Three corrections landed in lspace-det-sigma, and all three narrow claims rather than add results. The generated families had been deduplicated by diagram, so their counts were counts of diagrams: recounted by knot type, the 549 twisted torus records are 87 knots and the 14,452 random records are 46 — and every hyperbolic knot in those two families is already a SnapPy census knot, while the 274 one-bridge triples are 195 knots with at least 37 hyperbolic ones outside the census, so that family does extend the evidence (40cd61a). The claim that a counterexample in the sharp case could not be braid positive had no proof behind it and is withdrawn — a non-positive braid word does not make a knot non-braid-positive, and what has content is a set of twelve named knots, three of them provably not braid positive. The note now cites DeYeso’s thin L-space conjecture and Baldwin–Sivek’s Proposition 6.8 — an L-space knot with the Alexander polynomial of T(2,2g+1) is not a satellite — so a counterexample to the sharp case must be hyperbolic of genus at least 3, which is also what replaces the withdrawn property in PROMPTS.md (32f05ff, bd95a55). No violation appears and no conclusion reverses; the evidence beyond the census is thinner than the earlier version said.