""" Example cantor surfaces. """
from __future__ import annotations
import re
from typing import Iterable, Tuple
import bigger
from .utils import naturals, extract_curve_and_test
Edge = Tuple[int, int]
Link = Tuple[Edge, Edge, Edge, Edge]
[docs]def spotted_cantor() -> bigger.MCG[Edge]:
"""The uncountably-punctured sphere.
With mapping classes:
- a_n which twists about the curve across square n
- a{expr(n)} which twists about all a_n curves when expr(n) is True
Shortcuts:
- a[start:stop:step] = a{n in range(start, stop, step)}
- a == a[:]
"""
# #
# / \
# 2n+3,2 2n+4,2
# / \
# / \
# #---n,3---#
# | /|
# | / |
# n,1 n,0 n,1
# | / |
# |/ |
# #---n,2---#
def edges() -> Iterable[Edge]:
for x in naturals():
for y in range(4):
yield x, y
def link(edge: Edge) -> tuple[Edge, bool, Edge, bool, Edge, bool, Edge, bool]:
n, k = edge
# Down:
X, Y = (((n - 3) // 2, 3), (n + 1, 2)) if n % 2 == 1 else ((n - 1, 2), ((n - 3) // 2, 3))
# Three special down edges (squares 0, 1, & 2).
if k == 2 and 0 <= n < 3:
return ((n, 1), True, (n, 0), False, ((n + 2) % 3, 2), False, ((n + 1) % 3, 2), False)
return {
0: ((n, 3), False, (n, 1), False, (n, 2), True, (n, 1), True),
1: ((n, 0), False, (n, 2), True, (n, 0), True, (n, 3), False),
2: ((n, 1), True, (n, 0), False, X, True, Y, False),
3: ((2 * n + 4, 2), False, (2 * n + 3, 2), False, (n, 1), False, (n, 0), True),
}[k]
T = bigger.Triangulation.from_pos(edges, link)
def generator(name: str) -> bigger.Encoding[Edge]:
curve, test = extract_curve_and_test("a", name)
if curve == "a":
return T(lambda edge: 1 if edge[1] in {0, 1} and test(edge[0]) else 0).twist()
raise ValueError(f"Unknown mapping class {name}")
return bigger.MCG(T, generator)
[docs]def cantor() -> bigger.MCG[Edge]: # pylint: disable=too-many-statements
"""A sphere minus a cantor set.
With mapping classes:
- a_n which twists about the curve about the nth hole
- b_n which twists about the curve about the nth hole
- r an order two rotation
"""
POS, EQ, NEG = +1, 0, -1
def edges() -> Iterable[Edge]:
for x in naturals():
for y in [POS, EQ, NEG]:
yield x, y
def negate(X: Edge) -> Edge:
return X[0], -X[1]
def invert(sign: int, X: tuple[Edge, bool, Edge, bool, Edge, bool, Edge, bool]) -> tuple[Edge, bool, Edge, bool, Edge, bool, Edge, bool]:
return X if sign == POS else (negate(X[6]), not X[7], negate(X[4]), not X[5], negate(X[2]), not X[3], negate(X[0]), not X[1])
def link(edge: Edge) -> tuple[Edge, bool, Edge, bool, Edge, bool, Edge, bool]:
n, k = edge
if k == EQ: # Equator
if n == 0:
return ((0, NEG), False, (1, NEG), True, (1, POS), False, (0, POS), True)
elif n == 1:
return ((2, POS), False, (0, POS), False, (0, NEG), True, (2, NEG), True)
else: # n > 1
return ((3 * n - 3, NEG), False, (3 * n - 1, NEG), True, (3 * n - 1, POS), False, (3 * n - 3, POS), True)
# Northern / Southern hemisphere.
if n == 0:
return invert(k, ((0, EQ), False, (1, POS), False, (1, EQ), True, (2, POS), False))
elif n == 1:
return invert(k, ((4, POS), False, (3, POS), False, (0, POS), True, (0, EQ), False))
elif n == 2:
return invert(k, ((7, POS), False, (6, POS), False, (0, POS), False, (1, EQ), True))
N, r = n // 3 + 1, n % 3
incoming = 3 * (N // 2) - (1 if N % 2 else 2)
if r == 0:
return invert(k, ((N, EQ), False, (n + 2, POS), False, (incoming, POS), True, (n + 1, POS), False))
elif r == 1:
return invert(k, ((6 * N - 2, POS), False, (6 * N - 3, POS), False, (n - 1, POS), False, (incoming, POS), True))
else: # r == 2:
return invert(k, ((6 * N + 1, POS), False, (6 * N + 0, POS), False, (n - 2, POS), True, (N, EQ), False))
T = bigger.Triangulation.from_pos(edges, link)
def generator(name: str) -> bigger.Encoding[Edge]: # pylint: disable=too-many-branches
twist_match = re.match(r"(?P<curve>[ab])_(?P<n>-?\d+)$", name)
rotate_match = re.match(r"r$", name)
if twist_match is not None:
parameters = twist_match.groupdict()
curve_name = parameters["curve"]
N = int(parameters["n"])
if curve_name == "a":
if N == 1:
cut_sequence = [(0, EQ), (0, POS), (1, EQ)]
else:
cut_sequence = [(0, EQ), (N, EQ), (3 * N - 3, POS)]
while N > 1:
low_N = N // 2
cut_sequence.append((3 * low_N - (1 if N % 2 else 2), POS))
if N % 2:
cut_sequence.append((3 * low_N - 3, POS))
N = low_N
elif curve_name == "b":
if N <= 3:
cut_sequence = [(0, EQ), (0, POS), (1, EQ)]
else:
extend_left = N % 2
N = N // 2
cut_sequence = [(N, EQ), (3 * N - 3, POS)]
while N > 1:
N_low = N // 2
cut_sequence.append((3 * N_low - (1 if N % 2 else 2), POS))
if extend_left:
cut_sequence.append((3 * N_low - 3, POS))
if N % 2 != extend_left:
cut_sequence.append((N_low, EQ))
break
N = N_low
else:
cut_sequence.append((0, EQ))
curve = T(dict(((x, y * s), 1) for x, y in cut_sequence for s in [+1, -1]))
return curve.twist()
elif rotate_match is not None:
def isom(edge: Edge) -> Edge:
n, k = edge
if k == EQ:
if n == 0:
return (1, EQ)
elif n == 1:
return (0, EQ)
return (n ^ (1 << n.bit_length() - 2), k)
if n == 0:
return (0, k)
elif n == 1:
return (2, k)
elif n == 2:
return (1, k)
N, r = n // 3 + 1, n % 3
return (3 * (N ^ (1 << N.bit_length() - 2)) - 3 + r, k)
return T.encode([(-1, isom, isom)])
raise ValueError(f"Unknown mapping class {name}")
return bigger.MCG(T, generator)
