{"paper":{"title":"Implementation of the Habegger--Lin decision algorithm","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Implementation of the Habegger-Lin algorithm decides link-homotopy for four- and five-component links and finds new examples beyond Milnor invariants.","cross_cats":[],"primary_cat":"math.GT","authors_text":"Atsuhiko Mizusawa, Yuka Kotorii","submitted_at":"2026-05-06T02:53:44Z","abstract_excerpt":"Habegger and Lin gave a classification of link-homotopy classes of links in terms of that of string links modulo certain group actions. As an application, they constructed an algorithm for determining whether given two links are link-homotopic. In \\cite{KM4}, we explicitly computed these group actions for the 4- and 5-component cases. Consequently, the Habegger--Lin algorithm can be effectively applied in these cases. In this paper, we present an implementation of this algorithm, which is available at \\cite{KMcode}, and exhibit new pairs of links that are not link-homotopic yet cannot be disti"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we present an implementation of this algorithm, which is available at [KMcode], and exhibit new pairs of links that are not link-homotopic yet cannot be distinguished by Milnor's link-homotopy invariants, called μ-bar-invariants.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The explicit computations of the group actions on string links for the 4- and 5-component cases, reported in the authors' prior paper [KM4], are correct and complete.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Implementation of the Habegger-Lin decision algorithm for link-homotopy of 4- and 5-component links, together with new pairs not distinguished by Milnor invariants.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Implementation of the Habegger-Lin algorithm decides link-homotopy for four- and five-component links and finds new examples beyond Milnor invariants.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c7efbb159fc832f49618f7590be2e99a6bf40457499269fc585c32adf69f09ac"},"source":{"id":"2605.04433","kind":"arxiv","version":2},"verdict":{"id":"e3c80c4f-26bc-47fe-b48a-3495df7f72a1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T17:26:30.309736Z","strongest_claim":"we present an implementation of this algorithm, which is available at [KMcode], and exhibit new pairs of links that are not link-homotopic yet cannot be distinguished by Milnor's link-homotopy invariants, called μ-bar-invariants.","one_line_summary":"Implementation of the Habegger-Lin decision algorithm for link-homotopy of 4- and 5-component links, together with new pairs not distinguished by Milnor invariants.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The explicit computations of the group actions on string links for the 4- and 5-component cases, reported in the authors' prior paper [KM4], are correct and complete.","pith_extraction_headline":"Implementation of the Habegger-Lin algorithm decides link-homotopy for four- and five-component links and finds new examples beyond Milnor invariants."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.04433/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T23:01:19.943706Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:28:24.895085Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"4d2c6751ee2c24b1cce681b9fb53612a77a4ce16f20f30bff9a3ec87e1549a59"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","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"}