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Knots, Links, and 4-Manifolds

Ronald Fintushel, Ronald J. Stern

TL;DR

The paper establishes a deep link between knot/link invariants and the diffeomorphism types of 4-manifolds by treating Seiberg-Witten invariants as (multi)polynomials. It shows that fiber-summing along c-embedded tori multiplies SW invariants by Alexander polynomials, enabling every (monic) A-polynomial to arise as the SW invariant of a symplectic homotopy K3, while nonmonic polynomials yield nonsymplectic examples. It extends the knot/cusp framework to links, tying SW invariants to the multivariable Alexander polynomial and deriving a broad formula for the SW invariants of link-augmented manifolds. The work highlights how gauge-theoretic tools can generate vast families of exotic smooth structures on manifolds homeomorphic to K3 and related 4-manifolds, with a detailed treatment of the $b^+=1$ case and wall-crossing phenomena.

Abstract

In this paper we investigate the relationship between isotopy classes of knots and links in S^3 and the diffeomorphism types of homeomorphic smooth 4-manifolds. As a corollary of this initial investigation, we begin to uncover the surprisingly rich structure of diffeomorphism types of manifolds homeomorphic to the K3 surface.

Knots, Links, and 4-Manifolds

TL;DR

The paper establishes a deep link between knot/link invariants and the diffeomorphism types of 4-manifolds by treating Seiberg-Witten invariants as (multi)polynomials. It shows that fiber-summing along c-embedded tori multiplies SW invariants by Alexander polynomials, enabling every (monic) A-polynomial to arise as the SW invariant of a symplectic homotopy K3, while nonmonic polynomials yield nonsymplectic examples. It extends the knot/cusp framework to links, tying SW invariants to the multivariable Alexander polynomial and deriving a broad formula for the SW invariants of link-augmented manifolds. The work highlights how gauge-theoretic tools can generate vast families of exotic smooth structures on manifolds homeomorphic to K3 and related 4-manifolds, with a detailed treatment of the case and wall-crossing phenomena.

Abstract

In this paper we investigate the relationship between isotopy classes of knots and links in S^3 and the diffeomorphism types of homeomorphic smooth 4-manifolds. As a corollary of this initial investigation, we begin to uncover the surprisingly rich structure of diffeomorphism types of manifolds homeomorphic to the K3 surface.
Paper Structure (5 sections, 27 theorems, 84 equations)

This paper contains 5 sections, 27 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Background for the proofs of Theorems \ref{['knotthm']} and \ref{['linkthm']}
  3. The proof of Theorem \ref{['knotthm']}
  4. The proof of Theorem \ref{['linkthm']}
  5. Examples with $b^+=1$

Key Result

Theorem 1.1

Let $X$ be any simply connected smooth 4-manifold with $b^+>1$. Suppose that $X$ contains a smoothly c-embedded torus $T$ with $\pi_1(X\setminus T)=1$. Then for any $A$-polynomial $P(t)$, there is a smooth 4-manifold $X_{P}$ which is homeomorphic to $X$ and has Seiberg-Witten invariant where $t=\exp(2[T])$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • proof
  • Corollary 1.8
  • proof
  • ...and 25 more