Pith Number

pith:FDTTN4QF

pith:2026:FDTTN4QFAQIBQIJGHXXCOHSAGH

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CR-invariant energy of Legendrian knots in the Heisenberg group

Jun O'Hara, Yoshihiko Matsumoto

A CR-invariant energy for Legendrian knots in the Heisenberg group is minimized precisely by the R-circles.

arxiv:2604.25713 v2 · 2026-04-28 · math.GT · math.CV · math.DG

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Claims

C1strongest 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.

C2weakest 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).

C3one 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.

Receipt and verification

First computed	2026-05-20T01:05:14.592456Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

28e736f20504101821263dee271e4031d8eb2aa2d1792b59c3bb1aebeadbd818

Aliases

arxiv: 2604.25713 · arxiv_version: 2604.25713v2 · doi: 10.48550/arxiv.2604.25713 · pith_short_12: FDTTN4QFAQIB · pith_short_16: FDTTN4QFAQIBQIJG · pith_short_8: FDTTN4QF

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/FDTTN4QFAQIBQIJGHXXCOHSAGH \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 28e736f20504101821263dee271e4031d8eb2aa2d1792b59c3bb1aebeadbd818

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "2cf42e0999a4a297d1c826b2d5b21a46f0400ac7d482b735b97c2ab88226d91d",
    "cross_cats_sorted": [
      "math.CV",
      "math.DG"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.GT",
    "submitted_at": "2026-04-28T14:41:17Z",
    "title_canon_sha256": "d81b9f0c89df9e685b2e5b9c760ff073c1e31351d1a325efd8491c00eea312ca"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.25713",
    "kind": "arxiv",
    "version": 2
  }
}