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 Montesinos knot $K$ of length at least 3 has a persistently laminar branched surface unless it is equivalent to $K(1/2q_1,, 1/q_2,, 1/q_3,, -1)$forsomepositiveintegers$q_i$. 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.