Publication: On 2-universal knots
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2004
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Sociedad Matemática Mexicana
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Abstract
A link L is universal if every closed orientable 3-manifold M is a finite branched covering of S3 with the branch set equal to L. Known examples of universal links are the figure eight knot and the Borromean rings. It is also known that the trefoil knot is not universal.
The notion of universality can be refined by allowing only certain types of branching. A branched covering p:M→N over a link L is of type {1,2} if the branching (ramification) index of each component of p−1(K) is either 1 or 2. A link L is 2-universal if every closed orientable 3-manifold M is a finite branched cover of S3 over L, of type {1,2}.
The existence of 2-universal links has been known, but not of 2-universal knots. The authors use the existence of 2-universal links to prove that 2-universal knots exist. Their main theorem implies that for any 2-universal link L there exists a branched covering p:S3→S3 over a knot K, such that L is a sublink of the pseudo-branch cover, i.e., such that L is contained in the union of those components of p−1(K) which have branching indices 1. The link L is 2-universal; therefore for any given closed orientable 3-manifold M there exists a branched covering q:M→S3 over L, of type {1,2}. Furthermore, since L is a sublink of the pseudo-branch cover, the branched covering p∘q:M→S3 is also of type {1,2}, implying that K is 2-universal.
Part of the reason for studying 2-universal knots is the following. There exists a discrete universal group of hyperbolic isometries U (a discrete group of hyperbolic isometries G is universal if any closed orientable 3-manifold M is homeomorphic to the orbit space H3/H where H is a subgroup of G of finite index). The group U is generated by three 90∘ rotations. Since there are 3-manifolds which are not hyperbolic, any universal group has to contain rotations. The existence of a 2-universal hyperbolic knot would imply the existence of a discrete universal group of hyperbolic isometries that would only contain rotations by 180∘.
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