{"paper":{"title":"Quandle presentations of surface knots in 4-manifolds and bridge numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Banded unlink diagrams give Wirtinger presentations for the fundamental quandle of any surface link in a 4-manifold.","cross_cats":[],"primary_cat":"math.GT","authors_text":"Xiaozhou Zhou","submitted_at":"2026-05-14T09:02:26Z","abstract_excerpt":"The fundamental quandle is an invariant for distinguishing surface knots, yet computable presentations have traditionally been limited to surfaces embedded in the $4$-sphere. Building on the framework of banded unlink diagrams introduced by Hughes, Kim, and Miller, we give a Wirtinger type presentation of the fundamental quandle of surface links in arbitrary $4$-manifolds. As applications, we extend the work of Sato and Tanaka to show that for any $b \\geq 4$ and $m \\geq 0$, there exist infinitely many pairwise non-local surface knots with bridge number $b$ in $\\mathbb{C}P^2 \\#m\\overline{\\mathb"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we give a Wirtinger type presentation of the fundamental quandle of surface links in arbitrary 4-manifolds","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"that banded unlink diagrams extend directly to yield the quandle presentation in arbitrary 4-manifolds without additional topological restrictions","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A Wirtinger-type presentation for the fundamental quandle of surface links in arbitrary 4-manifolds is constructed, yielding infinite families of non-local knots with fixed bridge numbers in CP^2 # m CPbar^2.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Banded unlink diagrams give Wirtinger presentations for the fundamental quandle of any surface link in a 4-manifold.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"6731af69b9ff4bd9d201b2462cc26534e5b2fd55c284f4e6a0f2937ba423d8ed"},"source":{"id":"2605.14593","kind":"arxiv","version":1},"verdict":{"id":"b277cd86-1b99-488f-86f4-6df0109cbd0e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:14:04.818742Z","strongest_claim":"we give a Wirtinger type presentation of the fundamental quandle of surface links in arbitrary 4-manifolds","one_line_summary":"A Wirtinger-type presentation for the fundamental quandle of surface links in arbitrary 4-manifolds is constructed, yielding infinite families of non-local knots with fixed bridge numbers in CP^2 # m CPbar^2.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"that banded unlink diagrams extend directly to yield the quandle presentation in arbitrary 4-manifolds without additional topological restrictions","pith_extraction_headline":"Banded unlink diagrams give Wirtinger presentations for the fundamental quandle of any surface link in a 4-manifold."},"references":{"count":18,"sample":[{"doi":"","year":1965,"title":"Twisting spun knots , author=. Trans. Amer. Math. Soc. , volume=. 1965 , publisher=","work_id":"e1ccf852-8992-4424-a484-11d87643d82f","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"2-knots with the same knot group but different knot quandles , author=. J. Math. Soc. Japan , volume=. 2026 , publisher=","work_id":"5cc9f9dc-8e3e-4fd0-8d01-3dca723d4451","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Twist-spun torus knots , author=. Proc. Amer. Math. Soc. , volume=","work_id":"f06c2809-098d-47db-8453-a7f9d259d184","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Knots, groups, and , volume=","work_id":"99b8707a-fc57-4951-96a7-d4e7fcd03198","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1992,"title":"Racks and links in codimension two , author=. J. Knot Theory Ramifications , volume=. 1992 , publisher=","work_id":"cc0fe2f3-eb2f-4d07-b03a-3b2eb4ba2e2b","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":18,"snapshot_sha256":"4a47c5e6ba664032bd4a148b9731ae68ae8b3dad934c4e4ee41b606a1c069798","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}