In this paper, new examples of nonhomeomorphic knots and links which for certain r have homeomorphic r-sheeted cyclic branched coverings are constructed. In particular, it is proved that the two nonhomeomorphic knots with eleven crossings and with Alexander polynomial equal to one, have homeomorphic two-sheeted branched coverings, and that knots obtained from any knot by the Zeeman construction with p-fold and with q-fold twist have homeomorphic r-sheeted cyclic branched coverings ifp=±q(mod 2r). The construction of examples is based on regluing a link along a submanifold of codimension 1 by means of a homeomorphism which is covered by a homeomorphism which is isotopic to the identity only through nonprojecting isotopies.
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Viro, O.Y. Nonprojecting isotopies and knots with homeomorphic coverings. J Math Sci 12, 86–96 (1979). https://doi.org/10.1007/BF01098418
- Alexander Polynomial