Corks, exotic 4-manifolds and knot concordance

Abstract

We show that, for each integer n, there exist infinitely many pairs of n-framed knots representing homeomorphic but non-diffeomorphic (Stein) 4-manifolds, which are the simplest possible exotic 4-manifolds regarding handlebody structures. To produce these examples, we introduce a new description of cork twists and utilize satellite maps. As an application, we produce knots with the same 0-surgery which are not concordant for any orientations, disproving the Akbulut-Kirby conjecture given in 1978.

25 pages, 26 figures, Theorems 1.5 and 1.6 added, minor corrections

References in corpus (3)

• Stable isotopy in four dimensions
• Shake slice and shake concordant knots
• Fibered knots with the same $0$-surgery and the slice-ribbon conjecture

Cited by in corpus (9)

• Knot traces and concordance
• On the Upsilon invariant and satellite knots
• Corks with large shadow-complexity and exotic 4-manifolds
• From zero surgeries to candidates for exotic definite four-manifolds
• On Shake Slice Knots
• Nonexistence of twists and surgeries generating exotic 4-manifolds
• Twisted Mazur pattern satellite knots and bordered Floer theory
• $0$-Shake Slice Implies Slice
• Infinite nonabelian corks