def testInvolutiveRankL31(self):
"Check that the involutive Floer homology of L(3,1) is F_2^4."
P = infTypeD(1)
Q = infTypeD(1)
R = ArcslideDA(Arcslide(splitPMC(1),1,0))
Q2 = R.tensorD(Q)
Q3 = R.tensorD(Q2)
Q4 = R.tensorD(Q3)
othcx = P.morToD(Q4)
othcx.simplify()
cx = involutiveCx(P,Q3)
self.assertTrue(len(othcx)==3)
self.assertTrue(len(cx)==4)
def testGrading(self):
d1 = zeroTypeD(1)
d2 = infTypeD(1)
dd_id = identityDD(splitPMC(1))
cx = computeATensorD(d1.dual(), d2)
dstr1 = computeDATensorD(dd_id, d1)
dstr2 = computeATensorDD(d1, dd_id)
def testDATensorD(self):
dastr = identityDA(splitPMC(2))
dstr = zeroTypeD(2)
dstr_result = dastr.tensorD(dstr)
cx = dstr_result.morToD(infTypeD(2))
cx.simplify()
# Basic check that dstr_result is still zeroTypeD(2)
self.assertEqual(len(cx), 1)
def testMorToD(self):
id2 = identityDD(splitPMC(2))
d = infTypeD(2)
d2 = id2.morToD(d)
self.assertTrue(d2.testDelta())
d2.simplify()
self.assertEqual(len(d2), 1)
d3 = zeroTypeD(2)
d4 = id2.morToD(d3)
self.assertTrue(d4.testDelta())
d4.simplify()
self.assertEqual(len(d4), 1)
def openCap(self):
"""Returns the type D structure corresponding to this handlebody by
tensoring type DA bimodules with the genus-1 handlebody.
"""
assert self.genus > 0
last_move, prev_cap = self.getLastCobordism()
if self.genus == 1:
if last_move == 0:
return zeroTypeD(1)
else:
return infTypeD(1)
else:
cur_da = CobordismDALeft(Cobordism(self.genus, last_move, LEFT))
dstr = cur_da.tensorD(prev_cap.openCap())
dstr.simplify()
dstr.reindex()
return dstr
def platTypeD2(genus, is_dual = False):
"""Obtain linear handlebody from inf-framed handlebody by a sequence of
arcslides. As the inf-framed handlebody has absolute grading, we can get
absolute grading on this D structure.
"""
start = infTypeD(genus, is_dual)
slides = []
for i in range(genus-1):
slides += [(4*i+3,4*i+4), (4*i+6,4*i+7), (4*i+5,4*i+6)]
cur_pmc = splitPMC(genus)
for i in range(len(slides)):
b1, c1 = slides[i]
slides[i] = Arcslide(cur_pmc, b1, c1)
cur_pmc = slides[i].end_pmc
if not is_dual:
slides = [slide.inverse() for slide in slides]
slides_dd = [slide.getDDStructure() for slide in slides]
return composeDD(start, slides_dd, is_dual)
def testInvolutiveRankS3(self):
"Check that the involutive Floer homology of S^3 is F_2^2."
P = zeroTypeD(1)
Q = infTypeD(1)
cx = involutiveCx(P,Q)
self.assertTrue(len(cx)==2)