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.

Using Hirasawa-Murasugi's classification of fibered Montesinos knots we classify the L-space Montesinos knots, providing further evidence towards a conjecture of Lidman-Moore that L-space knots have no essential Conway spheres. In the process, we classify the fibered Montesinos knots whose open books support the tight contact structure on S 3 . We also construct L-space knots with arbitrarily large tunnel number and discuss the question of whether L-space knots admit essential tangle decompositions in the context of satellite operations and tunnel number.

The cosmetic crossing conjecture (also known as the "nugatory crossing conjecture") asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery characterization of the unknot to prove this conjecture for knots in integer homology spheres whose branched double covers are L-spaces satisfying a homological condition. This includes as a special case all alternating and quasi-alternating knots with square-free determinant. As an application, we prove the cosmetic crossing conjecture holds for all knots with at most nine crossings and provide new examples of knots, including pretzel knots, non-arborescent knots and symmetric unions for which the conjecture holds.homology sphere with rank y HF pY q " |H 1 pY ; Zq|, where y HF denotes the hat flavor of Heegaard Floer homology.Theorem 2. Let K be a knot in S 3 whose branched double cover ΣpKq is an L-space. If each summand of the first homology of ΣpKq has square-free order, then K satisfies the cosmetic crossing conjecture.Theorem 2 will be deduced from the Dehn surgery characterization of the unknot in L-spaces [Gai15, KMOS07] (see Theorem 11). It is interesting to juxtapose Theorem 2 with [BFKP12, Theorem 1.1], which implies that if a genus one knot K admits a cosmetic crossing change, then H 1 pΣpKqq is cyclic of order d 2 , for some d P Z.An abundant source of knots that meet the conditions of Theorem 2 are the Khovanov thin knots. These knots derive their definition from reduced Khovanov homology, which associates to an oriented link L in S 3 a bigraded vector spaceThe Ě Kh-thin links are those with their homology supported in a single diagonal δ " j´i of the bigradings. In this case, the dimension of Ě Kh is given by the determinant of the link. By work of Manolescu and Ozsváth [MO08], all quasi-alternating links are Ě Kh-thin, and this class includes all non-split alternating links [OS05]. Of relevance here is the fact that the branched double cover of a Ě Kh-thin link is an L-space, which follows from the spectral sequence from Ě KhpLq to y HF p´ΣpLqq [OS05] and the symmetry of Heegaard Floer homology under orientation reversal [OS04].Because the determinant of a knot is equal to the order of the first homology of its branched double cover we immediately obtain the following corollary. Corollary 3. A ĚKh-thin knot with square-free determinant satisfies the cosmetic crossing conjecture.

We prove that if the lens space L(n, 1) is obtained by a surgery along a knot in the lens space L(3, 1) that is distance one from the meridional slope, then n is in {−6, ±1, ±2, 3, 4, 7}. This result yields a classification of the coherent and noncoherent band surgeries from the trefoil to T (2, n) torus knots and links. The main result is proved by studying the behavior of the Heegaard Floer d-invariants under integral surgery along knots in L(3, 1). The classification of band surgeries between the trefoil and torus knots and links is motivated by local reconnection processes in nature, which are modeled as band surgeries. Of particular interest is the study of recombination on circular DNA molecules.1991 Mathematics Subject Classification. 57M25, 57M27, 57R58 (primary); 92E10 (secondary).