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These computations reveal some unexpected phenomena: we show that $s^\\sharp$ does not have to agree with $s$, and that $s^\\sharp$ is not additive under connected sums of knots.\n  Inspired by our computations, we separate the invariant $s^\\sharp$ into two "},"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":"1908.05018","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2019-08-14T08:45:32Z","cross_cats_sorted":[],"title_canon_sha256":"f018f7c86a25f4daf7acc90c66e2a28ae1a3e7badc26f18dc6942c4ba40d93ec","abstract_canon_sha256":"d08008f9ebc826eb6c922280922f473391843405d091adfd66a3e3b4afb6400f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:55:59.205499Z","signature_b64":"U2nQNi6gmAAUj7PxQFsZ4tgU43ZyIHctjuSMZzltDC2Sx77a01TrzHpWwg/ozJqMtyWPYP9JBqwh6c2505xUCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b8d704ee8b00649c5da467c514dd4540ad05eb3a9867f6778c2ba75fef41d85","last_reissued_at":"2026-07-04T23:55:59.205119Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:55:59.205119Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Kronheimer-Mrowka concordance invariant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Sherry Gong","submitted_at":"2019-08-14T08:45:32Z","abstract_excerpt":"Kronheimer and Mrowka introduced a new knot invariant, called $s^\\sharp$, which is a gauge theoretic analogue of Rasmussen's $s$ invariant. 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