ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 07 Nov 2017 16:15:51 -0600different results for Alexander polynomialshttp://ask.sagemath.org/question/39416/different-results-for-alexander-polynomials/ Hi! I just noticed that if one computes the Alexander polynomial of a knot, one gets different results according to whether the knot is presented as the closure of a braid or as a knot:
t = var('t')
B = BraidGroup(2)
b = B([1,1,1])
knot = Knot(b)
print b.alexander_polynomial()
print knot.alexander_polynomial()
Output: t^-2 - t^-1 + 1
t^-1 - 1 + t
Of course the polynomials differ only by a t^n multiplication, but I guess it would be better if the two coincided right away. My question is: is this issue going to be solved in the future? (I can live happily either way, just wanted to avoid modifying some stuff I'm working on!)
Thanks!
Tue, 07 Nov 2017 05:00:18 -0600http://ask.sagemath.org/question/39416/different-results-for-alexander-polynomials/Comment by dan_fulea for <p>Hi! I just noticed that if one computes the Alexander polynomial of a knot, one gets different results according to whether the knot is presented as the closure of a braid or as a knot:
t = var('t')
B = BraidGroup(2)
b = B([1,1,1])
knot = Knot(b)
print b.alexander_polynomial()
print knot.alexander_polynomial()</p>
<p>Output: t^-2 - t^-1 + 1
t^-1 - 1 + t
Of course the polynomials differ only by a t^n multiplication, but I guess it would be better if the two coincided right away. My question is: is this issue going to be solved in the future? (I can live happily either way, just wanted to avoid modifying some stuff I'm working on!)
Thanks! </p>
http://ask.sagemath.org/question/39416/different-results-for-alexander-polynomials/?comment=39434#post-id-39434Code:
t = var('t')
B = BraidGroup(2)
b = B( [1,1,1] )
k = Knot( b )
print b.alexander_polynomial( normalized=0 ) # 0 or 1...
print k.alexander_polynomial()
and `b.alexander_polynomial?` gives the answer
* "normalized" – boolean (default: "True"); whether to return the
normalized Alexander polynomial
OUTPUT:
The Alexander polynomial of the braid closure of the braid.
This is computed using the reduced Burau representation. The
unnormalized Alexander polynomial is a Laurent polynomial, which is
only well-defined up to multiplication by plus or minus times a power of t.
We normalize the polynomial by dividing by the largest power of t
and then if the resulting constant coefficient is negative
a.s.o .Tue, 07 Nov 2017 16:15:51 -0600http://ask.sagemath.org/question/39416/different-results-for-alexander-polynomials/?comment=39434#post-id-39434