Infinitely Many Knots Admitting the Same Integer Surgery and a Four-Dimensional Extension

Abstract

We prove that for any integer n there exist infinitely many different knots in S 3 such that n-surgery on those knots yields the same 3-manifold. In particular, when |n| = 1 homology spheres arise from these surgeries. This answers Problem 3.6(D) on the Kirby problem list. We construct two families of examples, the first by a method of twisting along an annulus and the second by a generalization of this procedure. The latter family also solves a stronger version of Problem 3.6(D), that for any integer n, there exist infinitely many mutually distinct knots such that 2-handle addition along each with framing n yields the same 4-manifold.2010 Mathematics Subject Classification. 57M25, 57M27, 57R65. 1 2 ABE, JONG, LUECKE, AND OSOINACH Here X K (n) denotes the smooth 4-manifold obtained from the 4-ball B 4 by attaching a 2-handle along K with framing n, and the symbol ≈ stands for a diffeomorphism.In Section 3, we generalize the annulus twist method in a somewhat surprising way to produce a different family of knots answering Problem 3.6(D). Furthermore, this family solves Problem 1.1 affirmatively as follows.Theorem 1.2. For every n ∈ Z, there exist distinct knots J 0 , J 1 , J 2 , . . . such thatThe knots J 0 and J 1 in Theorem 1.2 (for n > 0) are depicted in Figure 1, where the rectangle labelled n stands for n right-handed full twists. Note that J 0 is the knot 8 20 in Rolfsen's table [17]. The members of each infinite family are distinguished by their Alexander polynomials when n = 0. When n = 0, they are distinguished by hyperbolic volume (see [1]). J 0 J 1 1 1 n Figure 1. The knots J 0 and J 1 such that X J 0 (n) ≈ X J 1 (n). First family of knotsThe Dehn surgeries on a knot, K, in the 3-sphere are parameterized by their surgery slopes. These surgery slopes are described by p/q ∈ Q ∪ {∞}, meaning that the slope is a curve that runs p times meridionally and q times longitudinally (using the preferred longitude) along the boundary of the exterior of K. We write M K (p/q) for the p/q Dehn surgery on K. In this notation, an n-surgery on K refers to the integer surgery M K (n/1) = M K (n).Definition 2.1. Let L = k ∪ l 1 ∪ l 2 ∪ l 3 be the link pictured in Figure 2. Let L(α, β, δ, γ) be the corresponding Dehn surgery on L. Here the surgery slopes α, β, δ, γ will be either in Q ∪ {∞}, using the meridian-longitude coordinates on the boundary of a knot in S 3 (with

Fibered knots with the same $0$-surgery and the slice-ribbon conjecture

“…We outline the proof as follows: Let K 0 and K 1 be the unoriented knots depicted in Figure 1 and give arbitrary orientations on K 0 and K 1 . By using annular twisting techniques developed in [1,2,40,51], we see that 0surgeries on K 0 and K 1 give the same 3-manifold. On the other hand, by Miyazaki's result [36], we can prove that K 0 #K 1 is not ribbon, where K 0 #K 1 denotes the connected sum of K 0 and the mirror image of K 1 .…”

“…In this section, we prove our main theorem. The main tools are Miyazaki's result [36,Theorem 5.5] and annular twisting techniques developed in [1,2,40,51]. For the sake of completeness, we will review these results in Appendices A and B.…”

On the Stein framing number of a knot

“…To see this, observe that in the terminology of [AJLO14] the diagram on the left of Figure 2 is a simple annulus presentation for K n . Further, the knot K 0 n,t is obtained from K n by ( * t 2 )-twisting, which is defined by [AJLO14] and is a natural modification of the annulus twisting defined by [Oso06]. Therefore Theorem 3.10 of [AJLO14] implies that X t/2 (K n ) ∼ = X t/2 (K 0 n,t ).…”

“…Further, the knot K 0 n,t is obtained from K n by ( * t 2 )-twisting, which is defined by [AJLO14] and is a natural modification of the annulus twisting defined by [Oso06]. Therefore Theorem 3.10 of [AJLO14] implies that X t/2 (K n ) ∼ = X t/2 (K 0 n,t ). In the notation of the introduction, we take…”

“…The ideas for the proofs are as follows. For Theorem 1.2, we make use of work of Osoinach [Oso06], extended by Abe-Jong-Luecke-Osoinach [AJLO14], which gives a method to produce, for m ∈ N, pairs of distinct knots P m , Q m such that X −m (P m ) ∼ = X −m (Q m ). If X −m denotes this common 4-manifold, then X −m is Stein whenever −m is less than the maximal Thurston-Bennequin number of either P m or Q m .…”