Abstract. We give a method for obtaining infinitely many framed knots which represent a diffeomorphic 4-manifold. We also study a relationship between the n-shake genus and the 4-ball genus of a knot. Furthermore we give a construction of homotopy 4-spheres from a slice knot with unknotting number one.

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

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

No abstract

We give a family of linear inequalities which strictly estimate relation among the coefficients of the Alexander polynomials of alternating knots of genus two. We also give such families for positive knots of genus two, and for homogeneous knots of genus two. As an application, we determine the alternating knots of genus two such that the leading coefficients of them are less than or equal to three.