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My question is in the tittle:

Can we compute the Tristram–Levine signatures of a knot in $S^3$ using Jacobian with Fox partial derivatives?

If the answer is yes, is there a reference for this.

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I'm not sure exactly what you're allowing as input to this computation, but it can't be an arbitrary presentation of the knot group. For $K$ and $-K$ will have the same knot group, but they typically don't have the same signatures (eg for a trefoil knot). To put it another way, you'd have to take account of orientations somehow.

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  • $\begingroup$ How about a presentation obtained from a cell decomposition of $S^3\setminus \mathcal(K)$ ? This gives a matrix presentation of the first homology of the universal Abelian cover ? $\endgroup$ – christian Apr 9 '18 at 17:56
  • $\begingroup$ I guess I would make the same comment: $K$ and $-K$ (by which I mean the reflection of the knot) have the same homology of the universal Abelian cover. Perhaps the closest thing to what you are looking for is the interpretation of the Tristram-Levine signatures by Milnor in terms of the cup product structure on the infinite cyclic cover. See Milnor, Infinite cyclic coverings. $\endgroup$ – Danny Ruberman Apr 9 '18 at 19:44
  • $\begingroup$ Thanks, this is what I was thinking the cup product structure on the infinite cyclic coverings. $\endgroup$ – christian Apr 9 '18 at 20:54

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