{"paper":{"title":"Twisted homology jump loci, twisted Alexander polynomials, and $\\Sigma$-invariants","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GR","math.GT"],"primary_cat":"math.AG","authors_text":"Alexander I. Suciu, Yongqiang Liu","submitted_at":"2026-05-27T15:14:40Z","abstract_excerpt":"The twisted Alexander polynomials of a space, associated to a linear representation $\\sigma$ of the fundamental group, are non-abelian refinements of the classical Alexander polynomial from knot theory. In this paper, we show that they arise naturally from a new family of invariants -- the twisted homology jump loci -- which extend the rank-one characteristic varieties to higher-rank local systems. Using the tropical geometry of these twisted loci, we obtain sharper upper bounds for the Bieri--Neumann--Strebel--Renz (BNSR) $\\Sigma$-invariants.\n  For compact orientable $3$-manifolds with toroid"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.28595","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.28595/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}