I'm trying to use SnapPy [1] to calculate Alexander polynomials of knot
complements. SnapPy (which interfaces nicely with Sage) will happily
give me a presentation of the fundamental group of the knot complement,
and I want to take this and calculate the free derivatives of the group's
relators by the recursive formula
d(uv) = du + u * dv
d(u^-1) = -u^-1 * du
d(1) = 0
For a word w in the generators, we define the free derivative dw/dx to be
the coefficient of dx in the expression for dw - this will in general be
a polynomial in the (noncommuting) generators for the fundamental group.
I'm having a little difficult figuring out the best way to deal with
this in Sage - I need multivariate Laurent polynomials with noncommuting
variables - and wondered if anyone has any recommendations. I've been
trying to use a FreeAlgebra or FreeAlgebraQuotient but I'm not quite
sure how to go about this.
Any suggestions would be very welcome.
Nicholas
[1] http://www.math.uic.edu/~t3m/SnapPy/doc/
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