Double knot surgeries to S^4 and S^2 x S^2

TL;DR

This paper proves that performing knot surgeries along fishtails in S^4 and cusps in S^2 x S^2 does not alter their diffeomorphism types, regardless of the knots chosen.

Contribution

It establishes that knot surgeries along specific decompositions of S^4 and S^2 x S^2 are diffeomorphism invariants, extending understanding of these manifolds' structures.

Findings

Knot surgeries along fishtails in S^4 leave the manifold unchanged.

Knot surgeries along cusps in S^2 x S^2 do not alter the manifold.

The results hold for any choice of knots K and L.

Abstract

It is known that S^4 is a union of two fishtails, and S^2 x S^2 is a union of two cusps (glued along their boundaries). Here we prove that, for any choice of knots K,L in S^3, performing knot surgery operations to S^4 and S^2 x S^2, along both of the fishtails and cusps, respectively, do not change the diffeomorphism type of these manifolds.