different results for Alexander polynomials
 
  
 
  
 
 
 
 
 
 
    
        
        asked 7 years ago  
 
 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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Code:
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 .
dan_fulea gravatar imagedan_fulea (
    
        7 years ago
    )
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