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Two splittings from the same family become equivalent after at most one stabilization. If $K$ has bridge distance at least $4n$, then two splittings from different families become equivalent only after $n-1$ stabilizations. Further, we construct representatives of the isotopy classes of the minimal tunnel systems for $K$ corresponding to "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1507.07231","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-07-26T18:36:37Z","cross_cats_sorted":[],"title_canon_sha256":"ae5193b7fa19c84e4a0604b7df6a7346745413549034bc4f51c59ee6347c1376","abstract_canon_sha256":"74a30b577a5651be193b02e51337007446f57cd2055307939f63a322d239b6a5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:54:20.788986Z","signature_b64":"Ti30XYdPq47uZOzpyFtTiHxYg088ZT2QXTnCQR+weqinGgYZ/x5z82c9rmjUMdKREX7upzoDK2l4sOXWXF4iCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5db73113547463fcd1092555034fd111813717cbffccfe21fa88ab5ddee9925c","last_reissued_at":"2026-05-18T00:54:20.788588Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:54:20.788588Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stabilizing Heegaard Splittings of High-Distance Knots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"George Mossessian","submitted_at":"2015-07-26T18:36:37Z","abstract_excerpt":"Suppose $K$ is a knot in $S^3$ with bridge number $n$ and bridge distance greater than $2n$. 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