Persistently laminar branched surfaces

Abstract

We define sink marks for branched complexes and find conditions for them to determine a branched surface structure. These will be used to construct branched surfaces in knot and tangle complements. We will extend Delman's theorem and prove that a non-2-bridge Montesinos knot K has a persistently laminar branched surface unless it is equivalent to K(1/2q 1 , 1/q 2 , 1/q 3 , −1) for some positive integers q i . In most cases these branched surfaces are genuine, in which case K admits no atoroidal Seifert fibered surgery. It will also be shown that there are many persistently laminar tangles.

Complete exceptional surgeries on two-bridge links

“…Actually, he gave a construction of such an essential branched surface and describe them by using a combinatorial object called an allowable path. Based on the work of Li [10], Wu [14] proposed a sink mark description for branched surfaces. This description has made Delman's branched surfaces easy to treat.…”

“…In the following, we briefly review these studies for our purpose, that is, to study Dehn surgeries on two-bridge links. Our notations are basically the same used in [14], and we assume that the readers are somewhat familiar with those. For details about the definitions of terms used in the following, please refer [14] or [13,Section 5].…”

“…Our notations are basically the same used in [14], and we assume that the readers are somewhat familiar with those. For details about the definitions of terms used in the following, please refer [14] or [13,Section 5].…”

“…The construction of Σ was originally introduced by Delman [2]. Wu has reformulated the construction of Σ, and reprove that Σ is essential, see [14,Theorem 5.3]. Each channel creates two meridional cusps on ∂N(L) as in the proof of [14,Theorem 5.3].…”

“…Wu has reformulated the construction of Σ, and reprove that Σ is essential, see [14,Theorem 5.3]. Each channel creates two meridional cusps on ∂N(L) as in the proof of [14,Theorem 5.3]. In addition, one of the two meridional cusps is on V 1 , and the other is on V 2 as in the proof of [13,Lemma 5.3].…”