[Paper Review] Corks, exotic 4-manifolds and knot concordance

Kouichi Yasui|arXiv (Cornell University)|May 11, 2015

Geometric and Algebraic Topology26 references17 citations

TL;DR

This paper constructs infinitely many pairs of n-framed knots that yield homeomorphic but non-diffeomorphic Stein 4-manifolds for any integer n, using a new description of cork twists called 'hook surgery' and satellite maps. The method produces exotic smooth structures indistinguishable by Stein structures, and applies to disprove the 1978 Akbulut-Kirby conjecture on knot concordance by constructing knots with identical 0-surgeries that are not concordant for any orientation.

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.

Motivation & Objective

• To construct exotic 4-manifolds with minimal handlebody complexity, specifically n-framed knots yielding homeomorphic but non-diffeomorphic Stein 4-manifolds for any integer n.
• To develop a new description of cork twists—'hook surgery'—that enables systematic construction of such exotic pairs.
• To demonstrate that satellite maps can generate topologically equivalent but smoothly distinct 4-manifolds, revealing a fundamental difference between topological and smooth 4-manifold categories.
• To disprove the Akbulut-Kirby conjecture by constructing knots with identical 0-surgeries that are not concordant under any orientation.

Proposed method

• Introduce 'hook surgery' as a new description of cork twists, providing a geometric and algebraic framework to relate handlebody structures to smooth invariants.
• Utilize satellite maps P and Q to transform knots K satisfying specific conditions (2g₄(K) = ᾱd(K) + 2 and n ≤ tb̂(K)) into exotic n-framed knots Pₙ(K) and Qₙ(K).
• Construct pairs of satellite maps such that exchanging P and Q induces a cork twist, enabling the creation of exotic smooth structures.
• Apply the construction from Akbulut and Yasui (2015) to ensure both resulting 4-manifolds admit Stein structures when n ≤ tb̄(K) − 1.
• Use dot-zero surgery as a generalization of cork twists, showing that it can be described via hook surgeries under certain linking conditions.
• Define n-framed 4-dimensional satellite maps P⁽ⁿ⁾ and Q⁽ⁿ⁾, and prove they are topologically equivalent but smoothly distinct for infinitely many pairs.

Experimental results

Research questions

• RQ1 Can exotic 4-manifolds with minimal handlebody complexity be constructed for any framing n, beyond the known −1 and +1 cases?
• RQ2 Can a new geometric description of cork twists—hook surgery—be used to systematically generate exotic smooth structures?
• RQ3 Do satellite maps induce topologically equivalent but smoothly distinct 4-manifolds, revealing a categorical distinction in 4-dimensional topology?
• RQ4 Can the Akbulut-Kirby conjecture on knot concordance be disproven using 4-manifold invariants derived from surgery and satellite constructions?
• RQ5 Do all Mazur-type corks admit a hook surgery description, or are there exceptions?

Key findings

• For every integer n, there exist infinitely many distinct pairs of n-framed knots that yield homeomorphic but non-diffeomorphic 4-manifolds, with both manifolds admitting Stein structures.
• The construction relies on satellite maps Pₙ and Qₙ derived from hook surgery, which induce a cork twist when exchanged, producing exotic smooth structures.
• The 4-manifolds obtained from Pₙ(K) and Qₙ(K) are topologically equivalent but smoothly distinct, even when both admit Stein structures, showing that Stein structures cannot distinguish these exotic pairs.
• The paper disproves the Akbulut-Kirby conjecture by constructing knots with identical 0-surgeries that are not concordant for any orientation, using the new satellite map construction.
• A sufficient condition is given under which a dot-zero surgery is equivalent to a hook surgery, generalizing known exotic 4-manifold constructions.
• The construction yields infinitely many pairs of satellite maps that are topologically the same but smoothly distinct, demonstrating a new categorical divergence in 4-manifold topology.