def cap_off(matching, i):
"""
Merge i and i + 1 with a cap. Returns a new matching and whether
or not a circle was created.
>>> m = PerfectMatching([(0, 5), (1, 4), (2, 3)])
>>> cap_off(m, 2)
([(0, 3), (1, 2)], True)
>>> cap_off(m, 3)
([(0, 3), (1, 2)], False)
"""
def shift(a):
return a if a < i else a - 2
def follow(a):
b = matching.partner(a)
if b == i:
b = matching.partner(i + 1)
elif b == i + 1:
b = matching.partner(i)
return b
n = len(matching.base_set())
circle = matching.partner(i) == i + 1
new_match = set()
for a in range(n):
if not a in {i, i + 1}:
u, v = shift(a), shift(follow(a))
if u > v:
u, v = v, u
new_match.add((u, v))
return PerfectMatching(new_match), circle
def kauffman_bracket(link):
"""
>>> L = Link('T(2, 3)')
>>> kauffman_bracket(L)
q^-2 + 1 + q^2 - q^6
"""
ans = VElement()
if isinstance(link, exhaust.MorseEncoding):
encoded = link
else:
exhaustion = exhaust.MorseExhaustion(link)
encoded = exhaust.MorseEncoding(exhaustion)
for event in encoded:
if event.kind == 'cup':
ans = ans.insert_cup(event.min)
elif event.kind == 'cap':
ans = ans.cap_off(event.min)
else:
assert event.kind == 'cross'
if event.a < event.b:
ans = ans.add_positive_crossing(event.min)
else:
ans = ans.add_negative_crossing(event.min)
assert ans.is_multiple_of_empty_pairing()
return ans.dict[PerfectMatching([])]
def __init__(self, spec=None):
self.dict = dict()
if spec is None:
spec = PerfectMatching([])
if isinstance(spec, dict):
self.dict = spec
if isinstance(spec, PerfectMatching):
assert spec.is_noncrossing()
self.dict[spec] = R.one()
def insert_cup(matching, i):
"""
Insert a new adjacent matching which joins i and i + 1.
>>> m = PerfectMatching([(0, 1), (2, 5), (3, 4)])
>>> insert_cup(m, 0)
[(0, 1), (2, 3), (4, 7), (5, 6)]
>>> insert_cup(m, 1)
[(0, 3), (1, 2), (4, 7), (5, 6)]
>>> insert_cup(m, 6)
[(0, 1), (2, 5), (3, 4), (6, 7)]
"""
assert len(matching.base_set()) >= i
def shift(pair):
return [a if a < i else a + 2 for a in pair]
return PerfectMatching([shift(pair) for pair in matching] + [(i, i + 1)])
def is_multiple_of_empty_pairing(self):
return len(self.dict) == 1 and (PerfectMatching([]) in self.dict)