On cyclic branched coverings of prime knots

Abstract

We prove that a prime knot K is not determined by its p-fold cyclic branched cover for at most two odd primes p. Moreover, we show that for a given odd prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold cyclic branched cover of at most one more knot K' non equivalent to K. To prove the main theorem, a result concerning the symmetries of knots is also obtained. This latter result can be interpreted as a characterisation of the trivial knot.

28 pages, 2 figures

References in corpus (1)

• A characterisation of S^3 among homology spheres

Cited by in corpus (3)

• Quantum statistical mechanics in arithmetic topology
• Finite group actions and cyclic branched covers of knots in ${\mathbf{S}}^3$
• On the determination of knots by their cyclic unbranched coverings