Closures of $3$-braids and detection

TL;DR

The paper analyzes how three categorified link invariants—link Floer homology, Khovanov homology, and annular Khovanov homology—detect links arising from four closure operations on $3$-braids, leveraging classical braid classifications (Murasugi, Birman–Menasco, Baldwin–Sivek) to frame detection results. It proves that link Floer homology detects $L6a2$ and $L9n15$ and, for augmented clasp-closures, detects the Mazur link, while Khovanov homology detects $L6a2$ and $L9n15$, with Dowlin’s spectral sequence tying these results to knot Floer theory. The annular variant yields rank bounds tied to braid index via left-orderability, giving powerful detection results such as that $\mathrm{AKh}$ detects the Mazur pattern and certain $b(\sigma_1\sigma_2^{n})$ closures for $-2\le n\le 5$, with corresponding mod $2$ statements. A key technical contribution is a rank bound for annular Khovanov homology that informs both braid-closure and clasp-closure detection and enables new classification statements for closures of $3$-braids. Overall, the work advances detection capabilities for categorified link invariants in the specific but rich setting of $3$-braid closures and their closures, highlighting interplay between braid theory, low-dimensional topology, and homological invariants.

Abstract

We give some new link detection results for link Floer homology, Khovanov homology and annular Khovanov homology. The links we detect arise via different closure operations on $3$-braids. Examples of our results include that link Floer homology detects the Mazur link, that annular Khovanov homology detects the Mazur pattern, and that Khovanov homology detects L6a2 and L9n15. The Mazur pattern detection result depends on a new bound on the rank of the annular Khovanov homology of certain links.

Figures (4)

• Figure 1: The four types of links we study in this paper. The red dots indicates the axes.
• Figure 2: A sutured Heegaard diagram for the sutured manifold obtained by decomposing an augmented clasp-closure along a longitudinal surface for its axis. The outer boundary component of the surface corresponds to a longitude of $L_2$, while the inner boundary components correspond to meridians of $L_1$.
• Figure 3: The link $\mathring{c}(\sigma^n_1\sigma_2^{-1}\sigma_1\sigma_2)$. We consider the unoriented resolution of the crossing highlighted in orange.
• Figure 4: The resolutions for the canonical diagram --- as in Figure \ref{['fig:claspclosure']} --- for a clasp-closure yielding generators of $\mathop{\mathrm{CAKh}}\nolimits(c(\beta),k=-n;R)$. The solid red dot is the annular axis.

Theorems & Definitions (59)

• Remark 1
• Theorem 1.1
• proof : Proof of Theorem \ref{['thm:HFLL6a2']}
• Remark 1.2
• Theorem 1.3
• proof
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 1.7