Abstract
For any integer n > 2, the n-fold cyclic branched cover M of an alternating prime knot K in the 3-sphere determines K, meaning that if K is a knot in the 3-sphere that is not equivalent to K then its n-fold cyclic branched cover cannot be homeomorphic to M.
Highlights
A knot K in the 3-sphere is alternating if it admits a generic projection onto a 2sphere where the double points of the projection alternate between overcrossings and undercrossings when travelling along the knot
Before introducing the key ideas of the proof, it is worth mentioning what happens in the situations where the hypotheses of the theorem are not fulfilled, that is if the knots are not prime or if n = 2
This type of result cannot hold for composite knots
Summary
Abstract. — For any integer n > 2, the n-fold cyclic branched cover M of an alternating prime knot K in the 3-sphere determines K, meaning that if K is a knot in the 3-sphere that is not equivalent to K its n-fold cyclic branched cover cannot be homeomorphic to M. — For any integer n > 2, the n-fold cyclic branched cover M of an alternating prime knot K in the 3-sphere determines K, meaning that if K is a knot in the 3-sphere that is not equivalent to K its n-fold cyclic branched cover cannot be homeomorphic to M. — Pour tout entier n > 2, le revêtement ramifié cyclique à n feuillets M d’un nœud alterné K contenu dans la 3-sphère détermine K, à savoir si K est un nœud dans la 3-sphère qui n’est pas équivalent à K, alors son revêtement ramifié cyclique à n feuillets ne peut pas être homéomorphe à M Résumé. — Pour tout entier n > 2, le revêtement ramifié cyclique à n feuillets M d’un nœud alterné K contenu dans la 3-sphère détermine K, à savoir si K est un nœud dans la 3-sphère qui n’est pas équivalent à K, alors son revêtement ramifié cyclique à n feuillets ne peut pas être homéomorphe à M
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