Dehn surgeries and P2-reducible 3-manifolds

Spinal Knots in Lens Spaces

“…The problem is completely solved for torus knots [22] and satellite knots [2,14,24,25]. It is also known that there are many hyperbolic knots which produce lens spaces; among these the (−2, 3, 7)-pretzel knot [6] produces L (18,5) and L (19,7). The following result answers [4, Conjecture C].…”

“…2), which consists of a special subgraph in a disk on F . Generalized Scharlemann cycles appeared first in [5], where Σ is a projective plane, to prove that if G F contains a generalized Scharlemann cycle, then Σ is a non-minimal projective plane. Similar constructions are used by Hoffman [16] to prove that if Q is a minimal essential 2-sphere, then G F contains no generalized Scharlemann cycle (called closed cluster ), where Q takes the place of S Σ .…”

“…We follow the notations and definitions of [5]. Let Γ be the set of the simple closed curves bounding the white faces of Λ, and N be the number of Scharlemann cycles in D Λ .…”

P2-reducing and toroidal Dehn fillings

“…Then G P consists of at most two families of mutually parallel edges; one is a family of positive edges, and the other is that of negative edges. If k ≥ 3, then G P contains more than k mutually parallel negative edges by Lemma 5.1 (4). Then an easy Euler characteristic calculation shows that G K contains a 0-face and no level edges.…”

Homology of manifolds obtained by Dehn surgery on knots in lens spaces

“…[1], [3], [8], [9], [17]). This paper investigates the question for knots in lens spaces by means of homology calculations.…”