different results for Alexander polynomials

asked 2017-11-07 12:00:18 +0100

danieleC gravatar image

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!

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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


    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 .

dan_fulea gravatar imagedan_fulea ( 2017-11-07 23:15:51 +0100 )edit