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# Communications in Analysis and Geometry

## Volume 20 (2012)

### Number 2

### Persistently laminar branched surfaces

Pages: 397 – 434

DOI: http://dx.doi.org/10.4310/CAG.2012.v20.n2.a7

#### Author

#### 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.