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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

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