Pith Number

pith:NMM5YNXW

pith:2026:NMM5YNXW3KJSD5SRREYVBFTBX4

not attested not anchored not stored refs pending

Closed polylines with fixed self-intersection index

Dmitri Fomin

Closed polylines exist in which every one of the n edges is crossed exactly k times, for every k and all sufficiently large n making nk even.

arxiv:2605.05506 v3 · 2026-05-06 · math.MG

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{NMM5YNXW3KJSD5SRREYVBFTBX4}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We present a complete solution for k = 3 and k = 4, as well as the proof of some non-existence theorems. In conclusion, we show that, for an arbitrary positive integer k, a polyline of the required type exists for any sufficiently large integer n such that nk is even.

C2weakest assumption

The polylines are assumed to be closed chains in the Euclidean plane whose only intersections are transverse crossings, with the self-intersection index per edge being well-defined and independent of the particular embedding chosen.

C3one line summary

Complete solutions for uniform self-intersection index k=3 and k=4, plus a general existence theorem for sufficiently large n when nk is even.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:04:34.397023Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

6b19dc36f6da9321f6518931509661bf34a7ba93f1ef3954297ce04f1e10da3e

Aliases

arxiv: 2605.05506 · arxiv_version: 2605.05506v3 · doi: 10.48550/arxiv.2605.05506 · pith_short_12: NMM5YNXW3KJS · pith_short_16: NMM5YNXW3KJSD5SR · pith_short_8: NMM5YNXW

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/NMM5YNXW3KJSD5SRREYVBFTBX4 \
  | 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: 6b19dc36f6da9321f6518931509661bf34a7ba93f1ef3954297ce04f1e10da3e

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "99d13e7e593fb3be9316a7132c9cf587d02c679b316ca7a3a0b1c53a31faa376",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.MG",
    "submitted_at": "2026-05-06T23:06:12Z",
    "title_canon_sha256": "8cf97191c7af203ab03ed135ce1719910b71f0df76ca1239fb6d373a5a93a9bc"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.05506",
    "kind": "arxiv",
    "version": 3
  }
}