Cosmetic surgery in L-spaces and nugatory crossings

Abstract

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.

Cosmetic two‐strand twists on fibered knots

“…Suppose that |n| 2, and let K be the knot obtained from K by performing the two-strand n-twist determined by c. If K is isotopic to K and γ is unknotted in Σ(K), then the two-strand twist on K determined by c is nugatory. This is more general than the corresponding [18,Proposition 3.3]. We now review the proof given there, and provide the additional argument needed to prove the above proposition.…”

“…As stated in the introduction, we say that a two-strand twist on K is nugatory if a corresponding twisting circle c bounds a disk embedded in M − K. In this case, it follows readily that the lift of a corresponding twisting arc is unknotted in Σ(K), meaning that it bounds an embedded disk. A key fact for us, as in the work of [18,23], is that a partial converse holds true.…”

“…By the equivariant Dehn's lemma [5], we may assume that either τ ( Γ) ∩ Γ = ∅ or τ ( Γ) = Γ. As explained in [18], in the second case, Γ can in fact be perturbed to ensure that we are instead in the first case.…”

A note on band surgery and the signature of a knot

“…The statement of the lemma then follows. The reader may note that the proof of Lemma 2.2 below is adapted from the argument given in [17,Theorem 2.4]. In general, we will use the Smith normal form of the presentation matrix (2.2).…”

Symmetric Unions Without Cosmetic Crossing Changes

“…The infinite family of knots we construct here are shown in Section 3 to be non-alternating, non-fibered, hyperbolic, of genus two, and bridge number three. In a different direction, Theorem 3 was applied in [LM15] to settle the status of Conjecture 2 for all knots with up to nine crossings, families of pretzel knots of arbitrarily high genus, and certain knots arising as the branched sets of surgeries on strongly invertible L-space knots. In particular, the examples constructed in [LM15] were of non-constant determinant.…”