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.