Word hyperbolic Dehn surgery

Universal bounds for hyperbolic Dehn surgery

“…More recent work by Lackenby [33] and, independently, by Agol [2], similarly shows that for all but a universally bounded number of surgeries on each torus the resulting manifolds are irreducible with infinite word hyperbolic fundamental group. Similar types of bounds using techniques less comparable to those in this paper have been obtained by Gordon, Luecke, Wu, Culler, Shalen, Boyer, Zhang and many others.…”

Finite surgeries on three-tangle pretzel knots

“…2; p; q/ pretzel knot K [29], it follows from the work of Agol [1] and Lackenby [16] that any other exceptional slope s is within distance 10 of 2.p C q/.…”

“…This is equivalent to the minimal geometric intersection number of curves representing these two slopes. Agol [1] and Lackenby [16] independently showed that any pair of exceptional slopes on a one-cusped hyperbolic manifold lie within distance 10 of each other. Very recently, Lackenby and Meyerhoff [17] improved the bound from 10 to 8; we will not need this improvement.…”

“…In the first subsection, we use the 6-theorem of Agol [1] and Lackenby [16] to show that, in case 7 Ä p Ä q , the candidates for a finite slope of K are the trivial slope 1 0 and the integral slopes 2.p C q/ C k with k D 1; 0; 1; 2. Then, by Lemma 6, the boundary slopes [11] 2.p C q/ and 2.p C q/ C 2 are not finite slopes.…”

“…2; p; q/ pretzel knot with p , q odd and 5 Ä p Ä q . Then K admits no nontrivial finite surgery.Using the work of Agol [1] and Lackenby [16], candidates for finite surgery correspond to curves of length at most six in the maximal cusp of S 3 n K . If 7 Ä p Ä q , we will argue that only five slopes for the .…”