Pith Number

pith:ZCT4H5LC

pith:2013:ZCT4H5LCD5ANGOJ4B7XTX457JZ

not attested not anchored not stored refs pending

A note on the growth of Betti numbers and ranks of 3-manifold groups

Stefan Friedl

arxiv:1301.4025 v2 · 2013-01-17 · math.GT

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{ZCT4H5LCD5ANGOJ4B7XTX457JZ}

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.

Receipt and verification

First computed	2026-05-18T03:27:42.608292Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c8a7c3f5621f40d3393c0fef3bf3bf4e50fd00744ddb239cacf19f2aa17bb8cd

Aliases

arxiv: 1301.4025 · arxiv_version: 1301.4025v2 · doi: 10.48550/arxiv.1301.4025 · pith_short_12: ZCT4H5LCD5AN · pith_short_16: ZCT4H5LCD5ANGOJ4 · pith_short_8: ZCT4H5LC

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/ZCT4H5LCD5ANGOJ4B7XTX457JZ \
  | 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: c8a7c3f5621f40d3393c0fef3bf3bf4e50fd00744ddb239cacf19f2aa17bb8cd

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "64fefd112ef987fd0077747897e13deace08778061501fd7b8e049ff6deec8c2",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.GT",
    "submitted_at": "2013-01-17T09:28:36Z",
    "title_canon_sha256": "251d60498144513dff5451fd2e4534170744626e1d5a2afb002d6c0c016b3737"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1301.4025",
    "kind": "arxiv",
    "version": 2
  }
}