Immersed surfaces and Seifert fibered surgery on Montesinos knots

Abstract

We will use immersed surfaces to study Seifert fibered surgery on Montesinos knots, and show that if 1 q1−1 + 1 q2−1 + 1 q3−1 ≤ 1 then a Montesinos knot K( p1 q1 , p2 q2 , p3 q3 ) admits no atoroidal Seifert fibered surgery.

Persistently laminar branched surfaces

“…By a theorem of Brittenham [1], if K has a genuine persistently laminar branched surface then K(r) is not a small Seifert fibered manifold for any nontrivial r. Using this and the results of [12] we have the following two theorems, according to whether K is pretzel or not. Here a Montesinos knot is a pretzel knot if it can be written as K(1/q 1 , 1/q 2 , .…”

“…As in [12], we usep =p(p, q) to denote the mod q inverse of −p with minimal absolute value, i.e.,p satisfies pp ≡ −1 mod q, and 2|p| ≤ q. We can combine [12,Theorem 8.2] with Theorem 6.7 to obtain the following result for atoroidal Seifert fibered surgery on non-pretzel Montesinos knot. …”

Knots in handlebodies with nontrivial handlebody surgeries

“…We may therefore regard the pair (J, P ) as the exterior of a m/n rational tangle with P as the twice punctured disk on the left half sphere. By viewing the pair (J, P ) in this way, we will be able to apply the results of [13] regarding disks in J.…”

“…Let D be a compressing disk or a ∂-compressing disk whose boundary intersects ∂P minimally. Following [13], we say that D is an (r, s) disk if it meets ∂ 1 P ∪ ∂ 2 P in r points and ∂ 3 P in s points.…”

“…If D is a compressing disk for P , then it is a (0, 0) disk and [13,Lemma 2.3] implies that n = 0. If D is a ∂-compressing disk for P , then D is either a (1, 1) disk, a (2, 0) disk, or a (0, 2) disk.…”