Narrow Results

It is known that S⁴ is a union of two fishtails, and S² × S² is a union of two cusps (glued along their boundaries). Here we prove that for any choice of knots K, L ⊂ S³ the knot surgery operations and S² × S² ⇝ (S² × S²)_{K, L} along both of these fishtails and cusps, respectively, do not change the diffeomorphism type of these manifolds.

In this paper we clarify an issue in the knot surgery construction of Fintushel and Stern. Using knot surgery, they construct an infinite number of smooth structures on 4-manifolds satisfying certain conditions, but they do not explicitly work out the circumstances under which two manifolds that arise from their construction will fail to be diffeomorphic on the grounds of Seiberg–Witten theory. This paper fills in that gap.

Taking advantage of a numerical invariant, we visualize a distribution of rational homology 3-spheres on a plane via the Casson–Walker–Lescop (CWL) invariant and observe several aspects of the distribution. In particular, we study the characteristics of the distribution of lens spaces as a fundamental family of rational homology 3-spheres with a way to yield a family of estimation for the Dedekind sum. The CWL invariant captures the finiteness of lens space surgeries along knots. According to the finiteness, for example, the CWL invariant determines possible lens spaces as the results of integral surgeries along a knot $K$ with $a_1(K)=12$.