Floer homology and non-fibered knot detection

TL;DR

This work establishes that knot Floer homology and Khovanov homology detect non-fibered knots, and that HOMFLY homology detects infinitely many such knots, dramatically extending prior results restricted to fibered cases. The authors prove a main technical classification of genus-1 nearly fibered knots, then derive broad detection statements and Dehn-surgery applications, including infinite families characterized by 0-surgery and other slopes. The approach combines sutured Floer theory, essential annuli, and hyperelliptic involutions with a detailed 3-braid analysis and branched double-cover techniques to enumerate explicit knot types (notably $5_2$, $15n_{43522}$, certain Whitehead doubles, and pretzel knots). These results illuminate deep connections between Floer homologies, braids, and Dehn surgery, with implications for knot recognition and manifold topology. Overall, the paper provides a comprehensive framework for nearly fibered genus-1 knots and their detection by multiple homology theories, unlocking new pathways in knot theory and 3-manifold topology.

Abstract

We prove for the first time that knot Floer homology and Khovanov homology can detect non-fibered knots, and that HOMFLY homology detects infinitely many such knots; these theories were previously known to detect a mere six knots, all fibered. These results rely on our main technical theorem, which gives a complete classification of genus-1 knots in the 3-sphere whose knot Floer homology in the top Alexander grading is 2-dimensional. We discuss applications of this classification to problems in Dehn surgery which are carried out in two sequels. These include a proof that $0$-surgery characterizes infinitely many knots, generalizing results of Gabai from his 1987 resolution of the Property R Conjecture.

Figures (29)

• Figure 1: All of the genus-1 nearly fibered knots in $S^3$, up to taking mirrors; the labeled box on the right indicates the number of signed half-twists.
• Figure 2: Decomposing $(M_F,\gamma_F)$ along the annulus $A$ to form $(M_A,\gamma_A)$, and then removing the arc $\alpha$ to obtain the sutured manifold $(M_\alpha,\gamma_\alpha)$. The thick curves in the middle and right pictures indicate the sutures for these manifolds; there are no sutures on the left because $A(\gamma_F)$ is empty.
• Figure 3: We decompose $(M_A,\gamma_A)$ along $B$ to obtain $(M_2,\gamma_2)\sqcup (M',\gamma')$. Removing $\alpha$ and adding a meridional suture produces $(M_2,\gamma_2) \sqcup (M_3,\gamma_3)$, which is also the result of decomposing $(M_\alpha,\gamma_\alpha)$ along $B$.
• Figure 4: Left, the product sutured manifold $(M',\gamma')$, together with the arc $\alpha$. Right, the same manifold with $\alpha$ isotoped into $\partial M'$.
• Figure 5: Viewing $M_A$ as a submanifold of $M_F$, the arc $\alpha \subset \partial M_A$ lies in a push-off of the annulus $A$. On the right we see the region swept out by the isotopy of $\alpha$ into $A$.

Theorems & Definitions (123)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Remark 1.9
• Theorem \ref{thm:identify-y-c}