On hyperbolic knots with the same m-fold and n-fold cyclic branched coverings
✍ Scribed by Bruno Zimmermann
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 945 KB
- Volume
- 79
- Category
- Article
- ISSN
- 0166-8641
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✦ Synopsis
We have proved in previous work that, for any pair of different integers m > n > 2 (respectively m > n 3 2) which are not coprime, a hyperbolic (respectively 27r/n-hyperbolic) knot is determined by its m-fold and n-fold cyclic branched coverings; also, if TL is not a power of two, there exist at most two hyperbolic or 2x/n-hyperbolic knots with the same n-fold cyclic branched covering.
In the present paper, for any pair of coprime integers m. n. > 2, we construct the first examples of different hyperbolic knots having the same m-fold and also the same ?z-fold cyclic branched coverings; in fact there exist infinitely many different pairs of such knots. We construct also infinitely many triples of different z-hyperbolic knots such that the three knots of each triple have the same 2-fold branched covering; these coverings form an infinite series of hyperbolic homology 3-spheres starting from the spherical PoincarC homology 3-sphere. The question remains open how many different 7r-hyperbolic knots can have the same 2-fold branched covering (there are arbitrarily many hyperbolic knots with this property). 0 1997 Elsevier Science B.V.