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From knots to four-manifolds

Ciprian Manolescu

TL;DR

This survey analyzes the deep interplay between knot theory and four-dimensional topology through Kirby diagrams, which encode 4-manifolds as framed links, enabling knot-theoretic computations of smooth invariants.It highlights two principal frameworks—Heegaard Floer theory and skein lasagna modules—as robust tools for constructing and computing 4-manifold invariants, including exotic structures, often via surgery and cobordism maps.The article also outlines a program to study four-manifolds via knots on their boundaries, exploring sliceness, H-sliceness, and 0-friend phenomena as potential routes to new exotica and progress toward SPC4.Key contributions include combinatorial, algorithmic descriptions of Heegaard Floer invariants from link data, explicit exotic-pair examples derived from skein lasagna modules, and a structured program linking boundary knots to interior 4-manifold topology.Together, these developments provide a cohesive toolkit for translating 4-manifold questions into knot-theoretic data with practical computational avenues and potential breakthroughs in understanding smooth structures.

Abstract

This is a survey article about the connections between knot theory and four-dimensional topology. Every four-manifold can be represented in terms of a link, by a Kirby diagram. This point of view has led to progress in computing invariants of smooth four-manifolds that can detect exotic structures. We explain how this was done in two contexts: Heegaard Floer theory and skein lasagna modules. We also describe a program to understand four-manifolds through the properties of knots on their boundaries.

From knots to four-manifolds

TL;DR

This survey analyzes the deep interplay between knot theory and four-dimensional topology through Kirby diagrams, which encode 4-manifolds as framed links, enabling knot-theoretic computations of smooth invariants.It highlights two principal frameworks—Heegaard Floer theory and skein lasagna modules—as robust tools for constructing and computing 4-manifold invariants, including exotic structures, often via surgery and cobordism maps.The article also outlines a program to study four-manifolds via knots on their boundaries, exploring sliceness, H-sliceness, and 0-friend phenomena as potential routes to new exotica and progress toward SPC4.Key contributions include combinatorial, algorithmic descriptions of Heegaard Floer invariants from link data, explicit exotic-pair examples derived from skein lasagna modules, and a structured program linking boundary knots to interior 4-manifold topology.Together, these developments provide a cohesive toolkit for translating 4-manifold questions into knot-theoretic data with practical computational avenues and potential breakthroughs in understanding smooth structures.

Abstract

This is a survey article about the connections between knot theory and four-dimensional topology. Every four-manifold can be represented in terms of a link, by a Kirby diagram. This point of view has led to progress in computing invariants of smooth four-manifolds that can detect exotic structures. We explain how this was done in two contexts: Heegaard Floer theory and skein lasagna modules. We also describe a program to understand four-manifolds through the properties of knots on their boundaries.
Paper Structure (20 sections, 10 theorems, 52 equations, 19 figures)

This paper contains 20 sections, 10 theorems, 52 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Knot theory
  3. Four-manifolds
  4. Kirby diagrams
  5. Heegaard Floer theory
  6. Floer homologies.
  7. Heegaard Floer homology
  8. Knot Floer homology.
  9. Surgery formulas.
  10. Applications.
  11. Skein lasagna modules
  12. Khovanov homology.
  13. The skein module.
  14. A new invariant of four-manifolds.
  15. Computational methods.
  16. ...and 5 more sections

Key Result

Theorem 2.1

If a knot $K$ satisfies $\widehat{\mathit{HFK}}(K) \cong \widehat{\mathit{HFK}}(U)$, then $K$ is isotopic to the unknot $U$.

Figures (19)

  • Figure 1: (a) the unknot; (b) the Hopf link; (c) the trefoil; (d) the Borromean rings; (e) the Conway knot
  • Figure 2: Reidemeister moves
  • Figure 3: Three links that differ near a crossing
  • Figure 4: A diagram of the (right-handed) trefoil with its blackboard framing. This corresponds to $\lambda=3$.
  • Figure 5: A link diagram and its two resolutions at a crossing
  • ...and 14 more figures

Theorems & Definitions (17)

  • Theorem 2.1: Ozsváth-Szabó GenusBounds
  • Theorem 2.2: Kronheimer-Mrowka KMUnknot
  • Conjecture 3.1: Smooth four-dimensional Poincaré Conjecture (SPC4)
  • Theorem 4.1: Lickorish-Wallace lickorishWallace
  • Corollary 4.2
  • Theorem 5.1: MOS
  • Theorem 5.2: MOT
  • Remark 5.3
  • Theorem 5.4
  • Definition 6.1
  • ...and 7 more