A Family of Instanton-Invariants for Four-Manifolds and Their Relation to Khovanov Homology

TL;DR

This paper explores a family of instanton invariants for four-manifolds derived from gauge theory, establishing their connection to Khovanov homology and providing a framework for understanding knot invariants via four-dimensional gauge equations.

Contribution

It introduces a one-parameter family of Haydys-Witten instanton Floer homology groups and relates them to Khovanov homology through geometric and gauge-theoretic reductions.

Findings

Defined new instanton Floer homology groups $HF_{ heta}(W^4)$.

Established the relation between these groups and Khovanov homology for knot invariants.

Analyzed dimensional reductions of Haydys-Witten equations and boundary conditions.

Abstract

This article provides a review of the gauge-theoretic approach to Khovanov homology, framed in terms of a generalisation of Witten's original proposal. Concretely, the physical arguments underlying Witten's insights suggest that there is a one-parameter family of Haydys-Witten instanton Floer homology groups $HF_{\theta}\bigl(W^4\bigr)$ for four-manifolds. At the heart of the proposal is a systematic investigation of the dimensional reductions of the Haydys-Witten equations. It is shown that on the five-dimensional cylinder $M^5=\mathbb{R}_s\times W^4$ with nowhere-vanishing vector field $v=\cos\theta \partial_s+\sin\theta w$, the Haydys-Witten equations provide flow equations for the $\theta$-Kapustin-Witten equations on $W^4$. Similar reductions to lower dimensions include the twisted extended Bogomolny equations on three-manifolds and the twisted octonionic Nahm equations on one-manifolds, whose solutions provide natural boundary conditions along the boundary and corners of $W^4$. These reductions determine the indicial roots of the Haydys-Witten and $\theta$-Kapustin-Witten equations with twisted Nahm-pole boundary conditions, which are required to establish elliptic regularity. Motivated by these insights, the groups $HF_{\theta}\bigl(W^4\bigr)$ are defined in analogy with Yang-Mills instanton Floer theory: solutions of the $\theta$-Kapustin-Witten equations on $W^4$ modulo Haydys-Witten instantons on the cylinder $\mathbb{R}_s\times W^4$ interpolating between them. The relation to knot invariants observed by Witten arises when the four-manifold is the geometric blow-up $W^4=\bigl[X^3\times\mathbb{R}^+,K\bigr]$ along a knot $K\subset X^3\times{0}$ in its three-dimensional boundary. This yields a precise restatement of Witten's conjecture as the equality between $HF^\bullet_{\pi/2}\bigl(\bigl[S^3\times\mathbb{R}^+,K\bigr]\bigr)$ and Khovanov homology $\mathrm{Kh}^\bullet(K)$.