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Instantons and Khovanov homology in $\mathbb{RP}^3$

Hongjian Yang

TL;DR

This work develops a spectral-sequence bridge between Khovanov homology for links in $\mathbb{RP}^3$ and singular instanton Floer homology, and computes the $E_2$ pages for both null-homologous (class $0$) and nontrivial (class $1$) links. The authors show that the spectral sequence collapses to $\mathrm{I^{\sharp}}(\mathbb{RP}^3,K;\mathbb{Z}/2)$ (class $0$) or to the corresponding instanton invariant for class $1$, with a parallel story for reduced theories via $\mathrm{Kh}_1$. They prove that the reduced $\mathrm{Kh}$ over $\mathbb{Z}/2$ detects the unknot in class $0$ and the projective unknot in class $1$, using a rank inequality and results on instanton knot homology to certify the knot types. The paper also clarifies the structure of crossingless links in $\mathbb{RP}^3$ under instanton Floer theory, establishing canonical identifications up to signs and guiding potential extensions to broader detection problems in nontrivial 3-manifolds.

Abstract

We study the instanton Floer homology for links in $\mathbb{RP}^3$ and compute the second page of Kronheimer--Mrowka's spectral sequence. As an application, we show that Khovanov homology detects the unknot and the projective unknot in $\mathbb{RP}^3$.

Instantons and Khovanov homology in $\mathbb{RP}^3$

TL;DR

This work develops a spectral-sequence bridge between Khovanov homology for links in and singular instanton Floer homology, and computes the pages for both null-homologous (class ) and nontrivial (class ) links. The authors show that the spectral sequence collapses to (class ) or to the corresponding instanton invariant for class , with a parallel story for reduced theories via . They prove that the reduced over detects the unknot in class and the projective unknot in class , using a rank inequality and results on instanton knot homology to certify the knot types. The paper also clarifies the structure of crossingless links in under instanton Floer theory, establishing canonical identifications up to signs and guiding potential extensions to broader detection problems in nontrivial 3-manifolds.

Abstract

We study the instanton Floer homology for links in and compute the second page of Kronheimer--Mrowka's spectral sequence. As an application, we show that Khovanov homology detects the unknot and the projective unknot in .
Paper Structure (12 sections, 17 theorems, 84 equations, 3 figures)

This paper contains 12 sections, 17 theorems, 84 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Khovanov homology for links in $\mathbb{RP}^3$
  4. Instanton Floer homology
  5. Instanton homology of crossingless links in $\mathbb{RP}^3$
  6. The class $0$ unknot
  7. The class $1$ unknot
  8. Crossingless links and their isotopies
  9. Differentials on the $E_1$ page
  10. The class $0$ case
  11. The class $1$ case
  12. Detection results

Key Result

Theorem 1.1

Let $K$ be a knot in $\mathbb{RP}^3$. The reduced Khovanov homology $\widetilde{\mathop{\mathrm{Kh}}\nolimits}(K;\mathbb{Z}/2)$ has dimension $1$ if and only if $K$ is the unknot (when $K$ is of class $0$), or the projective unknot (when $K$ is of class $1$).

Figures (3)

  • Figure 1: Two types of smoothings.
  • Figure 2: A $1$-to-$1$ bifurcation. Notice that there is only one component on both sides as we are working on $\mathbb{RP}^2$, which is presented as a disk with antipodal points on the boundary attached. This can only happen when the link is in class $0$.
  • Figure 3: A bifurcation on $\mathbb{RP}^2$ with an essential circle. Notice that the diameter in the middle is a closed curve as we are woring on $RP^2$, and it represents the projective unknot $U'$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Lemma 2.1: manolescu2023rasmussen
  • Lemma 2.1: manolescu2023rasmussen
  • Theorem 2.2: manolescu2023rasmussen
  • Definition 2.3
  • Proposition 2.4
  • ...and 22 more