Networking Seifert Surgeries on Knots

Twist families of $L$-space knots, their genera, and Seifert surgeries

“…The exterior of c is a solid torus V = S 3 − N (c) containing K so that the manifold V (K; r) resulting from r-Dehn surgery on K in V has a Seifert fibration. If K(r) has a non-degenerate Seifert fibration, the main result of [44] (see also [12,Theorem 2.2]) shows that either r ∈ Z or K is a torus knot in V or a cable of a torus knot in V . Hence the situation when c is a seiferter for a (non-degenerate) Seifert surgery (K, r) with r ∈ Z is well understood.…”

“…We divide into two cases depending upon if the base orbifold is a disk with at least two cone points or a disk with at most one cone point. In the former case, K(r) is a connected sum of at least two lens spaces; in fact, [12,Proposition 2.8 (3)] shows that it has two lens space summands. It then follows from [23] that r ∈ Z.…”

“…In the course of the proof of Theorem 1.11, we characterize a twist family of surgeries which contains a large number of Seifert surgeries. In doing so, we extend the foundational work of [12] on seiferters. See Section 4 for terminology and background regarding seiferters and pseudo-seiferters.…”

“…Accordingly, a Seifert fibered space is a 3-manifold admitting a Seifert fibration with or without degenerate fibers. When we discuss surgeries, following the convention in [12], we call a knot-slope pair (K, r) a Seifert surgery if K(r) is a Seifert fibered space in our generalized sense. See Section 2 in [12] for degenerate Seifert fibrations.…”

“…When we discuss surgeries, following the convention in [12], we call a knot-slope pair (K, r) a Seifert surgery if K(r) is a Seifert fibered space in our generalized sense. See Section 2 in [12] for degenerate Seifert fibrations. Since connected sums of lens spaces are Seifert fibered L-spaces (in our sense), (T p,q , pq) is an L-space surgery and a Seifert surgery as well.…”