{"paper":{"title":"A module structure on odd Khovanov homology and the odd invariant for ribbon 2-knots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Jacob Migdail, Stephan Wehrli","submitted_at":"2026-07-04T20:28:12Z","abstract_excerpt":"We prove that the reduced odd Khovanov homology of a link $L$ is naturally a module over the exterior algebra of the first homology of the link's branched double-cover. We then describe this module structure more geometrically and related it to the odd Khovanov maps induced by link cobordisms. As an application, we will give a combinatorial proof of a recent result of Spyropoulos-Vidyarthi-Zhang about the odd invariant for $2$-knots in the special case where the $2$-knot is a ribbon $2$-knot. Additionally, we will show that Levine-Zemke's main result from their 2019 paper on Khovanov homology "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04018","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/2607.04018/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"}