Finite surgeries on three-tangle pretzel knots

We classify Dehn surgeries on (p, q, r) pretzel knots that result in a manifold of finite fundamental group. The only hyperbolic pretzel knots that admit non-trivial finite surgeries are (−2, 3, 7) and (−2, 3, 9). Agol and Lackenby’s 6-theorem reduces the argument to knots with small indices p, q, r. We treat these using the Culler-Shalen norm of the SL(2, C)-character variety. In particular, we introduce new techniques for demonstrating that boundary slopes are detected by the character variety. In [19] Mattman showed that if a hyperbolic (p, q, r) pretzel knot K admits a non-trivial finite Dehn surgery of slope s (i.e., a Dehn surgery that results in a manifold of finite fundamental group) then either • K = (−2, 3, 7) and s = 17, 18, or 19, • K = (−2, 3, 9) and s = 22 or 23, or • K = (−2, p, q) where p and q are odd and 5 ≤ p ≤ q. In the current paper we complete the classification by proving Theorem 1. Let K be a (−2, p, q) pretzel knot with p, q odd and 5 ≤ p ≤ q. Then K admits no non-trivial 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 \K. If 7 ≤ p ≤ q, we will argue that only five slopes for the (−2, p, q) pretzel knot have length six or less: the meridian and the four integral surgeries 2(p+ q) − 1, 2(p+ q), 2(p+ q) + 1, and 2(p+ q) + 2. If p = 5 and q ≥ 11, a similar argument leaves seven candidates, the meridian and the six integral slopes between 2(5 + q)− 2 and 2(5 + q) + 3. We will treat the remaining knots, (−2, 5, 5), (−2, 5, 7), and (−2, 5, 9), using the Culler-Shalen norm (for example, see [2, 5]). For a hyperbolic knot in S, this is a norm ‖ · ‖ on the vector space H1(∂M ;R). We can identify a Dehn surgery slope s ∈ Q ∪ { 1 0} with a class γs ∈ H1(∂M ;Z). If s is a finite slope that is not a boundary slope, the finite surgery theorem [2] shows that s is integral or half-integral and ‖γs‖ ≤ max{2S, S + 8} where S = min{‖γ‖ : 0 6= γ ∈ H1(M ;Z)} is the minimal norm. This makes the Culler-Shalen norm an effective tool for the study of finite surgery slopes. The Culler-Shalen norm is intimately related to the set of boundary slopes. An essential surface in the knot complement M will meet ∂M in a (possibly empty) set of parallel curves. The slope represented by this set of curves is known as a boundary slope. For a pretzel knot, these slopes are determined by the algorithm of Hatcher and Oertel [12]. Given the list of boundary classes {βj : 1 ≤ j ≤ N}, the norm is determined by an associated set of non-negative integers aj :