[Paper Review] Earrings, sutures and pointed links

Yi Xie|arXiv (Cornell University)|Sep 24, 2018

Geometric and Algebraic Topology11 references3 citations

TL;DR

This paper establishes a rank inequality between instanton knot homology and the Koszul homology of Khovanov homology for links in $S^3$, proving $2\dim_{\mathbb{C}}\operatorname{KHI}(L) \leq \operatorname{rank}_{\mathbb{Z}} H(K(\mathbf{X}, Kh(L)))$. The key innovation is constructing a spectral sequence from Baldwin-Levine-Sarkar's pointed Khovanov homology to a singular instanton invariant for pointed links, generalizing Kronheimer-Mrowka's result to links and preserving the Alexander grading via a new invariant $\operatorname{I}^{\sharp}(L,\mathbf{p})$. The result is sharp for unlinks and the Hopf link.

ABSTRACT

We prove a rank inequality on the instanton knot homology and the Khovanov homology of a link in $S^3$. The key step of the proof is to construct a spectral sequence relating Baldwin-Levine-Sarkar's pointed Khovanov homology to a singular instanton invariant for pointed links.

Motivation & Objective

• To extend Kronheimer and Mrowka's spectral sequence from knots to general links by defining a new singular instanton invariant for pointed links.
• To construct a spectral sequence from pointed Khovanov homology to a singular instanton invariant that preserves the Alexander grading.
• To prove a non-reduced rank inequality $2\dim_{\mathbb{C}}\operatorname{KHI}(L) \leq \operatorname{rank}_{\mathbb{Z}} H(K(\mathbf{X}, Kh(L)))$ for links in $S^3$, generalizing the knot case.
• To define $\operatorname{I}^{\sharp}(L,\mathbf{p})$ as a non-reduced version of $\operatorname{I}^{\natural}(L,\mathbf{p})$ that is isomorphic to $\operatorname{KHI}(L)$ and satisfies a spectral sequence to Khovanov homology.
• To show that the inequality is sharp for unlinks and the Hopf link, confirming its optimality in known cases.

Proposed method

• Define a pointed link $(L, \mathbf{p})$ with marking points on each component, and introduce the non-degenerate condition where each component has at least one marking.
• Construct $\operatorname{I}^{\sharp}(L,\mathbf{p})$ by adding an unknot with an earring to each marking point, generalizing $\operatorname{I}^{\natural}(L,\mathbf{p})$.
• Use Fukaya's connected sum theorem with $\mathbb{F}$-coefficients ($\operatorname{char} \neq 2$) to relate $\operatorname{I}^{\sharp}(L,\mathbf{p};\mathbb{F})$ to $\operatorname{I}^{\natural}(L,\mathbf{p};\mathbb{F})$ via a spectral sequence.
• Prove that the differential in the spectral sequence vanishes over $\mathbb{C}$, leading to $\dim_{\mathbb{C}}\operatorname{I}^{\sharp}(L,\mathbf{p};\mathbb{C}) = 2\dim_{\mathbb{C}}\operatorname{I}^{\natural}(L,\mathbf{p};\mathbb{C})$.
• Establish a spectral sequence from $Kh^\prime(L,\mathbf{p})$ to $\operatorname{I}^{\sharp}(L,\mathbf{p};\mathbb{C})$, using the unoriented skein exact triangle and iterated application of the triangle.
• Apply the spectral sequence and duality properties of Khovanov homology to derive the main inequality involving $\operatorname{KHI}(L)$ and the Koszul complex of $Kh(L)$.

Experimental results

Research questions

• RQ1 Can Kronheimer and Mrowka's spectral sequence from Khovanov homology to instanton homology be extended from knots to general links?
• RQ2 Does there exist a singular instanton invariant for pointed links that is isomorphic to $\operatorname{KHI}(L)$ and preserves the Alexander grading?
• RQ3 What is the relationship between the rank of the Koszul homology of $Kh(L)$ and the dimension of $\operatorname{KHI}(L)$ for links in $S^3$?
• RQ4 How does the new invariant $\operatorname{I}^{\sharp}(L,\mathbf{p})$ relate to $\operatorname{I}^{\natural}(L,\mathbf{p})$ and $\operatorname{KHI}(L)$ over $\mathbb{C}$?
• RQ5 Is the inequality $2\dim_{\mathbb{C}}\operatorname{KHI}(L) \leq \operatorname{rank}_{\mathbb{Z}} H(K(\mathbf{X}, Kh(L)))$ sharp for known link examples?

Key findings

• The paper proves $2\dim_{\mathbb{C}}\operatorname{KHI}(L) \leq \operatorname{rank}_{\mathbb{Z}} H(K(\mathbf{X}, Kh(L)))$ for any link $L$ in $S^3$, generalizing Kronheimer and Mrowka's knot result to links.
• For the unlink $U_m$ with $m$ components, $\dim_{\mathbb{C}}\operatorname{KHI}(U_m) = 2^{m-1}$ and $\operatorname{rank}_{\mathbb{Z}} H(K(\mathbf{X}, Kh(U_m))) = 2^m$, so the inequality is sharp.
• For the Hopf link $H$, $\dim_{\mathbb{C}}\operatorname{KHI}(H) = 4$ and $\operatorname{rank}_{\mathbb{Z}} H(K(\mathbf{X}, Kh(H))) = 8$, confirming the inequality is sharp in this case.
• The invariant $\operatorname{I}^{\sharp}(L,\mathbf{p};\mathbb{C})$ satisfies $\dim_{\mathbb{C}}\operatorname{I}^{\sharp}(L,\mathbf{p};\mathbb{C}) = 2\dim_{\mathbb{C}}\operatorname{I}^{\natural}(L,\mathbf{p};\mathbb{C})$, with the differential in the spectral sequence vanishing over $\mathbb{C}$.
• The spectral sequence from $Kh^\prime(L,\mathbf{p})$ to $\operatorname{I}^{\sharp}(L,\mathbf{p};\mathbb{C})$ is constructed via iterated application of the unoriented skein exact triangle, enabling the rank inequality.
• The reduced version of the inequality, $\dim_{\mathbb{C}}\operatorname{KHI}(L) \leq \operatorname{rank}_{\mathbb{Z}} H(K(\mathbf{X}^\prime, \widetilde{Kh}(L,p_0)))$, is also established, matching Kronheimer and Mrowka's original result for knots.