Double branched covers and pretzel knots

Abstract

Given a knot K we describe a modification of K which leaves the double branched cover of S 3 branched along K unchanged. We then modify certain pretzel knots in this way to produce arbitrarily large families of distinct knots having the property that all of the associated double branched covers are homeomorphic.1. Introduction. This paper concerns the relationship between a knot and its associated double branched cover. A brief review of the history of this problem will indicate some of the known results.Given This result can be interpreted in a different manner as follows. We note that the manifold constructed is a Seifert fibered manifold which then admits n distinct involutions where distinct here means non-conjugate in the automorphism group. For more on this see Plotnick [11].

Counting genus one fibered knots in lens spaces

“…For any given 3-manifold M , there may be several different links in S 3 with M as their double branched covers ( [2], [14], [1], among others). In particular, there may be several different links with representations as closed 3-braids that have M as their double branched covers.…”

Infinitely Many Knots with the Same Polynomial Invariant

“…Thus, using the pretzel knots [14], (cf. [1]), we have arbitrarily many distinct knots with the same polynomials.…”

Factorial growth rates for the number of hyperbolic 3-manifolds of a given volume

“…A proof similar to the one done by Boileau and Siebenmann can be found in [7,Theorem 12.29]. Also, Bedient gave a classification of a special class of Montesinos knots in [2]. and ∑ n j=1 1 q j ≤ n−2, are classified by the ordered set of fractions p 1 q 1 mod 1, .…”