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Suppose I am given a 3-manifold as a double branched cover over a link. Let a null-homologus knot in this space be given as a lift of an arc with endpoints on the link (it is automatically null-homologous if the double branched cover is a homology sphere, i.e. the link has determinant 1). Is there a way to calculate the Alexander polynomial of this knot simply from the link and the arc?

If not, can we say at least something about the Alexander polynomial, e.g. whether it is non-trivial, or what the determinant of the knot is?

Are there any conditions on the link and the arc that ensure that the Alexander polynomial is non-trivial?

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  • $\begingroup$ This brings up another question. There's a good chance that the double cover is not a homology sphere anymore. What do you mean by Alexander polynomial of a knot in this situation? One can think of quite a few abelian coverings that could be considered. $\endgroup$
    – Alex Degtyarev
    Jan 27, 2015 at 18:35
  • $\begingroup$ Thanks again! I meant that I want to consider only links that give rise to homology spheres. $\endgroup$
    – shestipalov
    Jan 27, 2015 at 18:59
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    $\begingroup$ Rather than asking that the double branched cover is a homology sphere, you could just first ask if the lifted knot is null-homologous. $\endgroup$
    – Ken Baker
    Jan 29, 2015 at 1:53
  • $\begingroup$ @KenBaker thanks, this is a good idea -- I edited the question accordingly $\endgroup$
    – shestipalov
    Feb 1, 2015 at 8:51

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