Cyclic and finite surgeries on Montesinos knots

Abstract

We give a complete classification of the Dehn surgeries on Montesinos knots which yield manifolds with cyclic or finite fundamental groups.

A GOOD PRESENTATION OF (-2, 3, 2s + 1)-TYPE PRETZEL KNOT GROUP AND ℝ-COVERED FOLIATION

“…Our Pretzel knots K s are in the class of a Montesinos knot. In [9], Ichihara and Jong showed that for a hyperbolic Montesinos knot K if K admits a non-trivial cyclic surgery it must be (−2, 3, 7)-pretzel knot and the surgery slope is 18 or 19, and if K admits a non-trivial acyclic finite surgery it must be (−2, 3, 7)-pretzel knot and the slope is 17, or (−2, 3, 9)-pretzel knot and the slope is 22 or 23. In contrast, by this theorem, infinitely many knots in the family of pretzel knot {K s } which appeared in Theorem 1.4 do not admit cyclic or finite surgery.…”

Surgery obstructions from Khovanov homology

“…, 31}). Finite fillings on this family of Montesinos links was left unresolved in Mattman's extensive study of the problem using character variety methods [30] (Mattman's classification has very recently been completed by Ichihara and Jong applying Heegaard Floer homology techniques [21], and independently treated by Futer, Ishikawa, Kabaya, Mattman and Shimokawa [13]). …”

Hyperbolic rational homology spheres not admitting fillable contact structures

“…It follows from Proposition 4.1 of[18] that the manifold obtained by any rational surgery r ≥ 9 on K is an L-space. So we show that M r,3 cannot bound a negative definite 4-manifold for any r ∈ (0,15]. Towards that end we use Theorem 2.2 and 2.3 of Owens and Strle.We compute the d invariant for the manifold obtained as 15-surgery on P (…”