{"paper":{"title":"Miyazawa's Invariant, Lefschetz Numbers, and Seifert Solids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Miyazawa's 2-knot invariant |deg| equals the Lefschetz number of a map on ordinary monopole Floer homology.","cross_cats":[],"primary_cat":"math.GT","authors_text":"Judson Kuhrman","submitted_at":"2026-05-14T15:56:44Z","abstract_excerpt":"We establish a formula expressing Miyazawa's 2-knot invariant $|\\mathrm{deg}|$ in terms of the Lefschetz number of a map on ordinary (i.e., not real) monopole Floer homology. As an application, we deduce that $|\\mathrm{deg}|=1$ for any 2-knot in $S^4$ which has a punctured $L$-space as a Seifert solid. In the course of the proof of the main theorem, we show how Francesco Lin's construction of monopole Floer homology with $\\operatorname{Pin}(2)$-equivariant perturbations can be made to work with integer coefficients."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish a formula expressing Miyazawa's 2-knot invariant |deg| in terms of the Lefschetz number of a map on ordinary (i.e., not real) monopole Floer homology.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That there exists a well-defined map on the monopole Floer homology whose Lefschetz number exactly recovers |deg|, and that Lin's Pin(2)-equivariant construction extends to integer coefficients without introducing new obstructions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Miyazawa's |deg| for 2-knots equals the Lefschetz number on ordinary monopole Floer homology, hence equals 1 for any 2-knot with a punctured L-space Seifert solid.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Miyazawa's 2-knot invariant |deg| equals the Lefschetz number of a map on ordinary monopole Floer homology.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f968a55e038ba75bc6d93d559e7196973dbb9d87d063f97c36219d1d872a95b6"},"source":{"id":"2605.14996","kind":"arxiv","version":1},"verdict":{"id":"ac62c801-be99-4fea-b932-7aa0d8dfa831","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T14:19:42.435790Z","strongest_claim":"We establish a formula expressing Miyazawa's 2-knot invariant |deg| in terms of the Lefschetz number of a map on ordinary (i.e., not real) monopole Floer homology.","one_line_summary":"Miyazawa's |deg| for 2-knots equals the Lefschetz number on ordinary monopole Floer homology, hence equals 1 for any 2-knot with a punctured L-space Seifert solid.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That there exists a well-defined map on the monopole Floer homology whose Lefschetz number exactly recovers |deg|, and that Lin's Pin(2)-equivariant construction extends to integer coefficients without introducing new obstructions.","pith_extraction_headline":"Miyazawa's 2-knot invariant |deg| equals the Lefschetz number of a map on ordinary monopole Floer homology."},"references":{"count":22,"sample":[{"doi":"10.2307/1970721","year":1968,"title":"Atiyah, M. F. and Bott, R. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1968 , PAGES =. doi:10.2307/1970721 , URL =","work_id":"da6c03ea-fe20-4aeb-83c5-64258dfd3b76","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/j.aim.2010.10.014","year":2011,"title":"Bloom, Jonathan M. , TITLE =. Adv. Math. , FJOURNAL =. 2011 , NUMBER =. doi:10.1016/j.aim.2010.10.014 , URL =","work_id":"cc7b9cb3-6d34-44c9-9e99-87e2dcb5836a","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1307/mmj/1523498585","year":2018,"title":"Michigan Math","work_id":"7cbe514f-b679-44fe-a929-c1479aaa6b4e","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.2307/1969592","year":1956,"title":"Fox, Ralph H. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1956 , PAGES =. doi:10.2307/1969592 , URL =","work_id":"41b18834-7592-4b7b-add1-84b71e5d2393","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1090/memo/1221","year":2018,"title":"Lin, Francesco , TITLE =. Mem. Amer. Math. Soc. , FJOURNAL =. 2018 , NUMBER =. doi:10.1090/memo/1221 , URL =","work_id":"8833aa0b-ce8f-4074-a8cc-e86ef88e7593","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":22,"snapshot_sha256":"3764d5586dc8f131a61389e87bf8a3830e19eb913a5860359d29b44902373067","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"08b03d678eabbaf14f3737bec8f6e840ad902c8710099a9cdbe988e549548d90"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}