[Paper Review] Null surgery on knots in L-spaces

Yi Ni, Faramarz Vafaee|arXiv (Cornell University)|Aug 25, 2016

Geometric and Algebraic Topology31 references3 citations

TL;DR

This paper proves that knots in L-spaces admitting Dehn surgery to a surface bundle over $S^1$ are rationally fibered, meaning their complements fiber over $S^1$. Using Heegaard Floer homology, the authors establish that such knots are Floer simple and that the induced contact structure on the ambient manifold is tight, generalizing classical fibered knot results from $S^3$ to $S^1 \times S^2$ and L-spaces.

ABSTRACT

Let $K$ be a knot in an L-space $Y$ with a Dehn surgery to a surface bundle over $S^1$. We prove that $K$ is rationally fibered, that is, the knot complement admits a fibration over $S^1$. As part of the proof, we show that if $K\subset Y$ has a Dehn surgery to $S^1 imes S^2$, then $K$ is rationally fibered. In the case that $K$ admits some $S^1 imes S^2$ surgery, $K$ is Floer simple, that is, the rank of $\hat{HFK}(Y,K)$ is equal to the order of $H_1(Y)$. By combining the latter two facts, we deduce that the induced contact structure on the ambient manifold $Y$ is tight. In a different direction, we show that if $K$ is a knot in an L-space $Y$, then any Thurston norm minimizing rational Seifert surface for $K$ extends to a Thurston norm minimizing surface in the manifold obtained by the null surgery on $K$ (i.e., the unique surgery on $K$ with $b_1>0$).

Motivation & Objective

• To generalize the fibered knot result from $S^3$ to $S^1 \times S^2$ and L-spaces.
• To establish that knots in L-spaces with $S^1 \times S^2$ surgeries are rationally fibered.
• To prove such knots are Floer simple and induce tight contact structures.
• To extend the theory of rational Seifert surfaces and Thurston norm minimality in null surgeries.
• To formulate and support a conjecture on fiberedness under the condition $[K]^\perp \subset \langle [K] \rangle$.

Proposed method

• Uses Heegaard Floer homology invariants, particularly $\widehat{HFK}(Y,K)$, to analyze knot simplicity and fiberedness.
• Applies the duality between $\widehat{HF}(Y)$ and $\widehat{HFK}(Y,K)$ to prove Floer simplicity when $K$ has an $S^1 \times S^2$ surgery.
• Employs the concept of semi-primitiveness in knot complements to relate homology classes and rational Seifert surfaces.
• Utilizes the Thurston norm minimality of rational Seifert surfaces and their extension after null surgery.
• Applies results from Cebanu and Rasmussen on fiberedness in lens spaces to generalize to L-spaces.
• Introduces and analyzes the condition $[K]^\perp \subset \langle [K] \rangle$ as a potential criterion for fiberedness.

Experimental results

Research questions

• RQ1 Are knots in L-spaces that admit $S^1 \times S^2$ surgeries necessarily rationally fibered?
• RQ2 Under what homological conditions does a Floer simple knot in an L-space become fibered?
• RQ3 Can the fiberedness of knots in $S^1 \times S^2$ with L-space surgeries be established without relying on classification of simple knots in lens spaces?
• RQ4 How do rational Seifert surfaces behave under null surgery in L-space knots?
• RQ5 Is there a generalized notion of positivity for spherical braids in $S^1 \times S^2$ corresponding to L-space surgeries?

Key findings

• Any knot $K$ in an L-space $Y$ with an $S^1 \times S^2$ surgery is rationally fibered, i.e., its complement fibers over $S^1$.
• Such knots are Floer simple, meaning $\text{rk }\widehat{HFK}(Y,K) = \text{rk }\widehat{HF}(Y)$.
• The induced contact structure on the ambient L-space $Y$ is tight, as a consequence of the knot being Floer simple and rationally fibered.
• Thurston norm minimizing rational Seifert surfaces for $K$ extend to norm-minimizing surfaces in the null surgery manifold.
• The condition $[K]^\perp \subset \langle [K] \rangle$ is both necessary and sufficient for the knot to be rationally fibered, supporting a conjecture on fiberedness in L-spaces.
• The proof generalizes Cebanu’s result from lens spaces to all L-spaces using Floer-theoretic techniques rather than case-by-case classification.