{"paper":{"title":"State integrals for the quantized $\\operatorname{SL}_2(\\mathbb{C})$ Chern-Simons invariant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.GT","authors_text":"Calvin McPhail-Snyder","submitted_at":"2026-01-08T17:28:08Z","abstract_excerpt":"Previous work of the author and N. Reshetikhin defines an invariant $\\operatorname{Z}_{N}^{\\psi}(K, \\rho, \\mu)$ of a knot $K$, a representation $\\rho : \\pi_{1}(S^{3} \\setminus K) \\to \\operatorname{SL}_2(\\mathbb{C})$, and a logarithm $\\mu$ of a meridian eigenvalue of $\\rho$. It can be interpreted as a geometric twist of the Kasahev invariant or as a quantization of the $\\operatorname{SL}(\\mathbb{C})$ Chern-Simons invariant and is defined using a discrete state-sum involving quantum dilogarithms. In this paper we show how to express $\\operatorname{Z}_{N}^{\\psi}(K, \\rho, \\mu)$ as a sum over conto"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.05136","kind":"arxiv","version":2},"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/2601.05136/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"}