  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 $K(1/2q_1,\,
1/q_2,\, 1/q_3,\, -1)$ 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.
