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A Seifert surgery is an integral surgery on a knot in S 3 producing a Seifert fiber space M which may contain an exceptional fiber of index 0. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; its edges correspond to single twistings along "seiferters" or "annular pairs of seiferters". One problem of the network is whether there is a path from each vertex to a vertex on a torus knot, the most basic Seifert surgery. We give a method to find seiferters and annular pairs of seiferters for Seifert surgeries obtained by taking two-fold branched covers of tangles. Concerning three infinite families of Seifert surgeries obtained by the second author via branched covers, we find explicit paths in the network from such surgeries to Seifert surgeries on torus knots. 1991 Mathematics Subject Classification. Primary 57M25, 57M50 Secondary 57N10.

A Seifert surgery is a pair (K , m) of a knot K in S 3 and an integer m such that m-Dehn surgery on K results in a Seifert fiber space allowed to contain fibers of index zero. Twisting K along a trivial knot called a seiferter for (K , m) yields Seifert surgeries. We study Seifert surgeries obtained from those on a trefoil knot by twisting along their seiferters. Although Seifert surgeries on a trefoil knot are the most basic ones, this family is rich in variety. For any m = −2 it contains a successive triple of Seifert surgeries (K , m), (K , m + 1), (K , m + 2) on a hyperbolic knot K , e.g. 17-, 18-, 19-surgeries on the (−2, 3, 7) pretzel knot. It contains infinitely many Seifert surgeries on strongly invertible hyperbolic knots none of which arises from the primitive/Seifertfibered construction, e.g. (−1)-surgery on the (3, −3, −3) pretzel knot.Keywords Dehn surgery · Hyperbolic knot · Seifert fiber space · Trefoil knot · seiferter · Seifert surgery network Dedicated to Fico González-Acuña on his 70th birthday.A. Deruelle

This paper concerns thin presentations of knots K in closed 3-manifolds M 3 which produce S 3 by Dehn surgery, for some slope γ . If M does not have a lens space as a connected summand, we first prove that all such thin presentations, with respect to any spine of M have only local maxima. If M is a lens space and K has an essential thin presentation with respect to a given standard spine (of lens space M ) with only local maxima, then we show that K is a 0-bridge or 1-bridge braid in M ; furthermore, we prove the minimal intersection between K and such spines to be at least three, and finally, if the core of the surgery K γ yields S 3 by r -Dehn surgery, then we prove the following inequality: |r| ≤ 2g, where g is the genus of K γ .