Pretzel knots with L-space surgeries

Abstract

Abstract. A rational homology sphere whose Heegaard Floer homology is the same as that of a lens space is called an L-space. We classify pretzel knots with any number of tangles which admit L-space surgeries. This rests on Gabai's classification of fibered pretzel links.

Floer homology and covering spaces

“…Proof. By [BM14,LM16b], the conditions on K guarantee that S 3 p/q (K) is not a Z/rZ-Lspace-knot. 1 On the other hand, S 3 p ′ /q ′ (P (−2, 3, 7)) is an L-space for p ′ /q ′ ≥ 9.…”

Mutations and short geodesics in hyperbolic 3-manifolds

“…Proof. All pretzel knots admitting lens space surgeries have been classified by Ichihara and Jong in [18], and this classification is also implied by the work of Lidman and Moore in [22]. Both works show that the only hyperbolic pretzel knot that admits any lens spaces surgeries is K 1 −2 , 1 3 , 1 7 .…”

Hyperbolic L-space knots and exceptional Dehn surgeries

“…To obtain an explicit picture of the covering knot K of κ, we apply isotopies given in Figs. [10][11][12][13][14][15]. Then taking the…”

“…Among alternating knots the only L-space knots are torus knots T 2n+1,2 [16]. Recent results of Lidman-Moore [10] and Baker-Moore [1] show that if K is a Montesinos L-space knot, then K is a pretzel knot P (−2, 3, 2n + 1) with n ≥ 0 (up to mirror image) or a torus knots T 2n+1,2 . Note that P (−2, 3, 2n + 1) with n ≥ 0 admits a Seifert fibered L-space surgery [10,16].…”

“…Recent results of Lidman-Moore [10] and Baker-Moore [1] show that if K is a Montesinos L-space knot, then K is a pretzel knot P (−2, 3, 2n + 1) with n ≥ 0 (up to mirror image) or a torus knots T 2n+1,2 . Note that P (−2, 3, 2n + 1) with n ≥ 0 admits a Seifert fibered L-space surgery [10,16]. One can find a large number of twist families of hyperbolic L-space knots each of which admits a Seifert fibered L-space surgery; see [14].…”