#17030: Knot Theory as a part of GSoC 2014.
-------------------------------------+-------------------------------------
Reporter: amitjamadagni | Owner: amitjamadagni
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-7.2
Component: algebraic | Resolution:
topology | Merged in:
Keywords: | Reviewers: Miguel Marco, Karl-
Authors: Amit Jamadagni, | Dieter Crisman, Frédéric Chapoton,
Miguel Marco | Travis Scrimshaw
Report Upstream: N/A | Work issues:
Branch: | Commit:
public/ticket/17030 | 0cce5891b429ea267c45bd89adacff6ebb12e453
Dependencies: | Stopgaps:
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Comment (by fuglede):
Cool, that's also what I suspected back then. Where I come from, Seifert
surfaces are connected by definition; that can make life a bit easier --
in particular in this situation where you can just use the same definition
for the Alexander polynomial regardless of the chosen surface.
In any case, by "cylinder", I meant something homeomorphic to ''S''^1^ ×
[0, 1] which has boundary two separated unknots, and which has non-trivial
''H'',,1,,. In that case, a Seifert matrix is the 1x1 zero matrix.
Likewise in the case of the unknot, you can avoid having to ponder about
0x0 matrices, since ''H'',,1,,(''D''^2^ × '''R''', '''Z''') may be
presented by the 1x1 unit matrix.
For the case at hand, the algorithm still produces incorrect results for
links which bound disconnected oriented surfaces (I would not call such
links "disjoint", since link components are ''always'' disjoint);
{{{
sage: B3 = BraidGroup(3)
sage: b = B3([1, 2, -2])
sage: b.alexander_polynomial()
0
sage: Link(b).alexander_polynomial()
1
}}}
And a weirder example:
{{{
sage: B2 = BraidGroup(2)
sage: b = B2([1, -1])
sage: b.alexander_polynomial()
0
sage: Link(b).alexander_polynomial()
---------------------------------------------------------------------------
ValueError Traceback (most recent call
last)
<ipython-input-14-c27fd62b3305> in <module>()
----> 1 Link(b).alexander_polynomial()
/home/fuglede/repos/sage/local/lib/python2.7/site-
packages/sage/knots/link.pyc in alexander_polynomial(self, var)
1244 R = LaurentPolynomialRing(ZZ, var)
1245 # The Alexander polynomial of disjoint links are defined
to be 0
-> 1246 if len(self._braid_word_components()) > 1:
1247 return R.zero()
1248 t = R.gen()
/home/fuglede/repos/sage/local/lib/python2.7/site-
packages/sage/knots/link.pyc in _braid_word_components(self)
899 ml = list(self.braid().Tietze())
900 if not ml:
--> 901 raise ValueError("the braid remains the same with no
components")
902
903 l = set(abs(k) for k in ml)
ValueError: the braid remains the same with no components
}}}
--
Ticket URL: <http://trac.sagemath.org/ticket/17030#comment:170>
Sage <http://www.sagemath.org>
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