We give some new examples of smooth 4-manifolds which are diffeomorphic although they are obtained by Fintushel-Stern knot surgeries on a smooth 4-manifold with different knots the first such examples are given by Akbulut. In the proof we essentially use the monodromy of a cusp. | Turk J Math 30 (2006) , 87 – 93. ¨ ITAK ˙ c TUB A Connected Sum of Knots and Fintushel-Stern Knot Surgery on 4-manifolds Manabu Akaho Abstract We give some new examples of smooth 4-manifolds which are diffeomorphic although they are obtained by Fintushel-Stern knot surgeries on a smooth 4-manifold with different knots; the first such examples are given by Akbulut [1]. In the proof we essentially use the monodromy of a cusp. 1. Introduction Let X be a smooth 4-manifold. In [4] a cusp in X is a PL embedded 2-sphere of selfintersection 0 with a single nonlocally flat point whose neighborhood is the cone on the right-hand trefoil knot. The regular neighborhood of a cusp is called a cusp neighborhood. It is fibered by smooth tori with one singular fiber, the cusp, and the monodromy is 1 1 −1 0 ! . If T is a smoothly embedded torus representing a nontrivial homology class [T ], we say that T is c-embedded if T is a smooth fiber in a cusp neighborhood. Consider an oriented knot K in S 3 , and let m denote an oriented meridional circle to K; see Figure 1. Let MK be the 3-manifold obtained by performing 0-framed surgery on K. Then m can also be viewed as a circle in MK . In MK × S 1 we have a smooth torus 1991 Mathematics Subject Classification: Primary 57R55. Secondary 57M25 Supported by JSPS Grant-in-Aid for Scientific Research (Wakate (B)) 87 AKAHO m K Figure 1 Tm = m × S 1 of self-intersection 0. Since a tubular neighborhood of m has a canonical framing in MK , a tubular neighborhood of the torus Tm in MK × S 1 has a canonical identification with Tm × D2 . Let X(K,φ) denote the fiber sum X(K,φ) := [X \ (T × D2 )] ∪φ [(MK × S 1 ) \ (Tm × D2 )], where T × D2 is a tubular neighborhood of the torus T in the manifold X and φ : ∂(T × D2 ) → ∂(Tm × D2 ) is a homeomorphism. In general, the diffeomorphism type of X(K,φ) depends on φ. If we fix an identification of T with S 1 × S 1 and a homeomorphism φ : ∂(T × D2 ) → ∂(Tm × D2 ) such that φ(S 1 × ∗ × ∗) = m × ∗ ×
