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Twisted Alexander polynomials of periodic knots. (English) Zbl 1097.57010
Authors’ abstract: K. Murasugi [Comment. Math. Helv. 46, 162–174 (1971; Zbl 0206.25603)] discovered two criteria that must be satisfied by the Alexander polynomial of a periodic knot. We generalize these to the case of twisted Alexander polynomials. Examples demonstrate the application of these new criteria, including to knots with trivial Alexander polynomial, such as the two polynomial 1 knots with 11 crossings. R. Hartley [Can. J. Math. 33, 91–102 (1981; Zbl 0481.57003)] found a restrictive condition satisfied by the Alexander polynomial of any freely periodic knot. We generalize this result to the twisted Alexander polynomial and illustrate the applicability of this extension in cases in which Hartley’s criterion does not apply.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57M27 Invariants of knots and $$3$$-manifolds (MSC2010)
##### Software:
KnotInfo; Knotscape; SnapPea
Full Text:
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