Persistently laminar branched surfaces

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(1/2q1, 1/q2, 1/q3, −1) for some positive integers qi. 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.