r"""Volumes of the orientable cusped SnapPea census manifolds -- numberdb.org/T225 Vol(M), the volume of the complete finite-volume hyperbolic metric of curvature -1, for the orientable cusped hyperbolic 3-manifolds M in the m part of SnapPy's OrientableCuspedCensus. Run it with SageMath: $ sage -pip install numberdb snappy $ sage -python generate.py $ sage -python generate.py --publish The values are computed by SnapPy's verified volume method, which returns Sage intervals containing the true volumes. The table stores the 301 census names beginning with m, which is the original SnapPea census through five ideal tetrahedra. """ import os import sys import numberdb.sage as numberdb from sage.rings.real_arb import RealBallField from snappy import Manifold, OrientableCuspedCensus WORKING_GUARD = 64 EXPECTED_COUNT = 301 SPECIAL_COMMENTS = { "m003": ( "Its volume is $2\\,\\mathrm{Cl}_2(\\pi/3)$, twice " "HREF{Values_of_the_Clausen_functions_at_rational_multiples_of#2,1/3}" "[Gieseking's constant]." ), "m004": ( "This is the figure-eight knot complement, " "HREF{Hyperbolic_volumes_of_the_prime_knots_with_at_most_ten_crossings#4,1}" "[the $4_1$ row in the table of prime-knot complement volumes]. " "Its volume is $2\\,\\mathrm{Cl}_2(\\pi/3)$, twice " "HREF{Values_of_the_Clausen_functions_at_rational_multiples_of#2,1/3}" "[Gieseking's constant]." ), "m129": ( "This is the Whitehead link complement, the link $5^2_1$ of " "Rolfsen's table. Its volume is $4G=4\\,\\mathrm{Cl}_2(\\pi/2)$, " "where $G$ is " "HREF{Values_of_the_Clausen_functions_at_rational_multiples_of#2,1/2}" "[Catalan's constant]." ), } def _key_from_stdin(): if os.environ.get("NUMBERDB_KEY_FROM_STDIN") != "1": return token = sys.stdin.read().strip() if "=" in token and token.split("=", 1)[0].isupper(): token = token.split("=", 1)[1].strip().strip("'\"") if token: os.environ["NUMBERDB_API_KEY"] = token def census_names(): names = [] seen = set() for manifold in OrientableCuspedCensus: name = manifold.name().split("(", 1)[0] if not name.startswith("m") or name in seen: continue seen.add(name) names.append(name) if len(names) != EXPECTED_COUNT: raise ArithmeticError( "found %d m-census names, expected %d" % (len(names), EXPECTED_COUNT) ) return names def homology_tex(manifold): text = str(manifold.homology()) text = text.replace("Z", "\\mathbb{Z}") text = text.replace(" + ", "\\oplus") return text def comment(name): manifold = Manifold(name) cusp_word = "cusp" if manifold.num_cusps() == 1 else "cusps" tetra_word = "tetrahedron" if manifold.num_tetrahedra() == 1 else "tetrahedra" base = ( "It has %d %s, an ideal triangulation with %d %s, and first " "homology $%s$." % ( manifold.num_cusps(), cusp_word, manifold.num_tetrahedra(), tetra_word, homology_tex(manifold), ) ) special = SPECIAL_COMMENTS.get(name) if special: return "%s %s" % (base, special) return base def verified_volume(name, digits): bits = numberdb.bits(digits, losing=WORKING_GUARD) return Manifold(name).volume(verified=True, bits_prec=bits) def decimal_enclosure(text, RB): mantissa = text.lower().split("e", 1)[0] if "." not in mantissa: return RB(text) places = len(mantissa.split(".", 1)[1]) radius = RB(10) ** (-places) return RB(text).add_error(radius) def number_from_nested(table, *keys): node = table["Numbers"] for key in keys: node = node[key] return node["number"] def check_identities(digits=100): bits = numberdb.bits(digits, losing=WORKING_GUARD) RB = RealBallField(bits) widest = None for name in census_names(): interval = verified_volume(name, digits) approximate = decimal_enclosure(str(Manifold(name).high_precision().volume()), RB) difference = RB(interval) - approximate if not difference.contains_zero(): raise ArithmeticError( "%s: verified volume does not contain high-precision volume: %s" % (name, difference) ) width = interval.absolute_diameter() if widest is None or width > widest[0]: widest = (width, name, interval) knot_table = numberdb.table("T141") bianchi_table = numberdb.table("T221") m004 = RB(verified_volume("m004", digits)) figure_eight = decimal_enclosure(number_from_nested(knot_table, "4", "1"), RB) if not (m004 - figure_eight).contains_zero(): raise ArithmeticError("m004 does not overlap the stored figure-eight volume") m009 = RB(verified_volume("m009", digits)) m010 = RB(verified_volume("m010", digits)) if not (m009 - m010).contains_zero(): raise ArithmeticError("m009 and m010 do not overlap") bianchi_d7 = decimal_enclosure(number_from_nested(bianchi_table, "-7"), RB) if not (m009 - 3 * bianchi_d7).contains_zero(): raise ArithmeticError("m009 is not three times the D=-7 Bianchi covolume") m129 = RB(verified_volume("m129", digits)) bianchi_d4 = decimal_enclosure(number_from_nested(bianchi_table, "-4"), RB) if not (m129 - 12 * bianchi_d4).contains_zero(): raise ArithmeticError("m129 is not twelve times the D=-4 Bianchi covolume") print("checked %d m-census manifolds" % EXPECTED_COUNT) print("widest verified volume at %s: %s" % (widest[1], widest[2])) print("m004 agrees with T141 4_1") print("m009 and m010 agree, and m009 = 3 * Bianchi D=-7") print("m129 = 12 * Bianchi D=-4") class OrientableCuspedSnapPeaVolumes(numberdb.Generator): table = os.environ.get("NUMBERDB_TABLE") or "T225" parameters = ("name",) type = "R" digits = 100 rigour = "proven" files = ("generate.py",) def enumerate(self): for name in census_names(): yield {"name": name} def value(self, params, digits): name = str(params["name"]) return {"number": verified_volume(name, digits), "comment": comment(name)} def fill_draft_once(generator, message): """Fill a fresh draft without the empty upsert probe.""" from numberdb._generate import ( _check_precision, _check_rigour, _producer, _run_name, _source_files, ) from numberdb._write import Entries, attach, submit_entries, to_text table = generator.table run = _run_name(generator) entries = Entries(*generator.parameters) for params in generator.enumerate(): params = dict(params) wanted = generator.digits_for(params) entry = generator._entry(params, wanted) value = entry["number"] identity = ",".join(str(params[name]) for name in generator.parameters) _check_rigour(generator, table, identity, value) written = to_text(value, wanted, generator.format) _check_precision(table, identity, written, wanted, lowering=False) record = dict(entry) record.pop("digits", None) entries.add(**params, **record, digits=wanted) answer = submit_entries( table, entries, message=message, produced_by=_producer(generator), upsert=False, run=run, rigour=generator.rigour, ) files = _source_files(generator) stored = [] for name, body in sorted(files.items()): attach(table, name, body, run=run, message=message, rigour=generator.rigour) stored.append(name) return { "tid": answer.get("tid", table), "revision": answer.get("revision"), "entries": len(entries), "files": stored, } if __name__ == "__main__": _key_from_stdin() generator = OrientableCuspedSnapPeaVolumes() if os.environ.get("NUMBERDB_CHECK_IDENTITIES") == "1": check_identities(generator.digits) elif "--publish" in sys.argv or os.environ.get("NUMBERDB_PUBLISH") == "1": print(fill_draft_once( generator, message="orientable cusped SnapPea census volumes")) else: report = generator.verify(sample=None) print(report) sys.exit(0 if report.ok else 1)