{"paper":{"title":"CR-invariant energy of Legendrian knots in the Heisenberg group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A CR-invariant energy for Legendrian knots in the Heisenberg group is minimized precisely by the R-circles.","cross_cats":["math.CV","math.DG"],"primary_cat":"math.GT","authors_text":"Jun O'Hara, Yoshihiko Matsumoto","submitted_at":"2026-04-28T14:41:17Z","abstract_excerpt":"We introduce an energy functional for Legendrian knots in the 3-dimensional Heisenberg group $\\mathcal{H}$, which serves as a sub-Riemannian analog of the M\\\"obius invariant knot energy in Euclidean 3-space introduced by the second author. The energy is obtained by regularizing a divergent integral of the potential of order -2 with respect to the Kor\\'anyi distance on $\\mathcal{H}$; this choice of distance is essential for the energy to be invariant under the action of PU(2,1). We characterize $\\mathbb{R}$-circles in $\\mathcal{H}$ as the minimizers of the energy, and establish a Heisenberg ana"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We characterize R-circles in H as the minimizers of the energy, and establish a Heisenberg analog of the Doyle-Schramm cosine formula. We also show that the energy integrand admits an expression in terms of a complex-valued 2-form on the complement of the diagonal in H×H.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The regularization of the divergent integral of the order -2 potential with respect to the Koranyi distance yields a finite, CR-invariant functional whose minimizers are precisely the R-circles; this depends on the specific choice of distance being essential for invariance under PU(2,1).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The authors introduce a CR-invariant energy for Legendrian knots in the Heisenberg group, prove that R-circles minimize it, and derive a Heisenberg analog of the Doyle-Schramm cosine formula together with a complex 2-form expression for the integrand.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A CR-invariant energy for Legendrian knots in the Heisenberg group is minimized precisely by the R-circles.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9d818faa9ce59d749f95315b28d8406097d05b77a83d7515ea40eb07910afe9b"},"source":{"id":"2604.25713","kind":"arxiv","version":2},"verdict":{"id":"e16f2911-f705-4a14-9312-01fa20290735","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T14:13:26.094069Z","strongest_claim":"We characterize R-circles in H as the minimizers of the energy, and establish a Heisenberg analog of the Doyle-Schramm cosine formula. We also show that the energy integrand admits an expression in terms of a complex-valued 2-form on the complement of the diagonal in H×H.","one_line_summary":"The authors introduce a CR-invariant energy for Legendrian knots in the Heisenberg group, prove that R-circles minimize it, and derive a Heisenberg analog of the Doyle-Schramm cosine formula together with a complex 2-form expression for the integrand.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The regularization of the divergent integral of the order -2 potential with respect to the Koranyi distance yields a finite, CR-invariant functional whose minimizers are precisely the R-circles; this depends on the specific choice of distance being essential for invariance under PU(2,1).","pith_extraction_headline":"A CR-invariant energy for Legendrian knots in the Heisenberg group is minimized precisely by the R-circles."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.25713/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T20:51:32.935595Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"a618d6b69f48dfc172633e78446cd3ffe3084f9066b9460980ad55c1193af1d1"},"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"}