#17030: Knot Theory as a part of GSoC 2014.
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Reporter: amitjamadagni | Owner: amitjamadagni
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-7.0
Component: algebraic | Resolution:
topology | Merged in:
Keywords: | Reviewers: Miguel Marco, Karl-
Authors: Amit Jamadagni, | Dieter Crisman, Frédéric Chapoton
Miguel Marco | Work issues:
Report Upstream: N/A | Commit:
Branch: | 79d175ab20b41008e74ab0895b4781d7358b6dd5
public/ticket/17030 | Stopgaps:
Dependencies: |
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Comment (by tscrim):
Linear algebra question: What is the determinant of a 0x0 matrix? FYI -
The Sage answer:
{{{
sage: mat = matrix(ZZ, 0, 0)
sage: mat.det()
1
}}}
This is why the Alexander polynomial is returning 1 for the example in
comment:142. IMO, I think it should be 0 because the sum in the definition
of the determinant is vacuous. (A related question, is a 0x0 matrix
invertible?)
I get the genus is 0 for the example in comment:142 with the current
version.
I'm starting to understand what is going on with the code. The homology
generators is setting a value of 0 if a particular pair `(L[i], L[i+1])`
in the braid word component vector does not contribute to a homology
generator. The braid word components are a nice reduced expression to
compute these generators. See Lemma 3.1 in
http://www.maths.ed.ac.uk/~jcollins/SeifertMatrix/SeifertMatrix.pdf.
I'm going through right now and doing some code improvements, seeing if I
can simplify things, and working around the issue noted above.
--
Ticket URL: <http://trac.sagemath.org/ticket/17030#comment:155>
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