Nicholas,
Its possible that Axiom's XPOLY non-commutative polynomial domain
might be of some help. E.g.
sage: x=axiom('x::XDistributedPolynomial(Symbol,Integer)')
sage: y=axiom('y::XDistributedPolynomial(Symbol,Integer)')
sage: z=axiom('z::XDistributedPolynomial(Symbol,Integer)')
sage: p=2*x*y*z
sage: q=z*x*y+3*z
sage: pq=p*q
sage: pq
2 2
6x y z + 2x y z x y
sage: pq.leadingCoefficient()
2
sage: pq.leadingMonomial()
2
x y z x y
sage: pq.reductum()
2
6x y z
But note that XPOLY does not allow negative exponents and you would
have to build the derivative operations.
Implementing non-commutative Laurent polynomials in Axiom/FriCAS would
not be difficult (There is already a commutative Laurent polynomial
domain which can serve as a model.) Adding free derivatives would be
a little more involved. If you are at all interested, please let me
know.
In any case, XPOLY might serve as a starting point for something
similar in Sage.
Regards,
Bill Page.
On Thu, Aug 13, 2009 at 11:39 AM, Nicholas
Jackson<nicholas.jack...@warwick.ac.uk> wrote:
>
> 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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