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Math Help - Alexander Polynomial

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    MHF Contributor Swlabr's Avatar
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    Alexander Polynomial

    Given a knot, we can construct a group presentation, and from this presentation we can get an Alexander Polynomial. It is a knot invariant.

    My question is this: Given an arbitrary (finite) presentation which maps onto \mathbb{Z} by setting all the generators to be equal, do we get an Alexander Polynomial? Further, if there are more relators than generators, is the polynomail always zero? (I would guess Yes for the first, No for the second, but I think I read somewhere that you take it to be zero if there are more relators than regenerators, which contradicted something the author was saying elsewhere...thus my question!)

    For example, the presentation \langle a, b, c, x, y; ax=yb, ay=xb, bx=xa, by=yc, cx=xc, cy=ya\rangle has zero Alexander polynomial, while \langle a, b, c; ab=ca, bc=ab, ca=bc \rangle has Alexander Polynomail t^2-t+1.
    Last edited by Swlabr; December 14th 2010 at 09:04 AM.
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