{"paper":{"title":"Lefschetz Fibrations on Knot Traces of Alternating and Extended Alternating Knots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Knot traces of alternating and extended alternating knots admit positive allowable Lefschetz fibrations whose regular fibers have genus equal to the number of white regions in the associated planar graph.","cross_cats":[],"primary_cat":"math.GT","authors_text":"Atsushi Tanaka","submitted_at":"2026-05-13T05:15:24Z","abstract_excerpt":"In our previous work, we introduced a simple and explicit method for constructing a positive allowable Lefschetz fibration (PALF) from a $2$-handlebody decomposition of any given compact Stein surface. In this paper, we apply this construction to knot traces whose attaching circles are either alternating knots or \\emph{extended alternating knots} (a generalized class introduced herein). We demonstrate that each such knot trace admits a PALF whose regular fiber has a genus exactly equal to the number of white regions in the associated planar graph, yielding PALFs whose regular fibers have a sig"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Each such knot trace admits a PALF whose regular fiber has a genus exactly equal to the number of white regions in the associated planar graph.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The prior construction method from the authors' previous work applies without additional obstructions to the 2-handlebody decompositions of these specific knot traces.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Knot traces of alternating knots and the newly defined extended alternating knots admit PALFs whose regular fibers have genus equal to the number of white regions in their planar graphs.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Knot traces of alternating and extended alternating knots admit positive allowable Lefschetz fibrations whose regular fibers have genus equal to the number of white regions in the associated planar graph.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"8f00abda47cf10c8c76bde6bf8b2a44651d12418986864ae96ac97931899899f"},"source":{"id":"2605.13017","kind":"arxiv","version":1},"verdict":{"id":"642fc5e9-517a-428b-b6d1-7acf33d4d9d3","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T02:04:18.520358Z","strongest_claim":"Each such knot trace admits a PALF whose regular fiber has a genus exactly equal to the number of white regions in the associated planar graph.","one_line_summary":"Knot traces of alternating knots and the newly defined extended alternating knots admit PALFs whose regular fibers have genus equal to the number of white regions in their planar graphs.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The prior construction method from the authors' previous work applies without additional obstructions to the 2-handlebody decompositions of these specific knot traces.","pith_extraction_headline":"Knot traces of alternating and extended alternating knots admit positive allowable Lefschetz fibrations whose regular fibers have genus equal to the number of white regions in the associated planar graph."},"references":{"count":24,"sample":[{"doi":"10.1142/s0219199712500356","year":2012,"title":"Firat , TITLE =","work_id":"a7e50abd-b820-4e2e-af7f-cb5de2f17830","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.2140/gt.2001.5.319","year":2001,"title":"Akbulut, Selman and Ozbagci, Burak , TITLE =. 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