Surgeries on knots and tight contact structures

TL;DR

The paper addresses the problem of when Dehn surgeries along knots in $S^3$ carry tight contact structures detectable by Heegaard Floer invariants. It blends Heegaard Floer theory, the LOSS invariant for rationally null-homologous Legendrian knots, the dual knot surgery mapping cone, and sutured Floer theory to control contact invariants under rational and negative surgeries. Key contributions include an injectivity result at the top Alexander grading, naturality of LOSS and contact invariants under surgery, and a construction of admissible Legendrian representatives with nonzero LOSS to realize tight structures after surgery; the main result is that for any knot $K$ and any $r>0$, $S^3_{-r}(K)$ has a tight contact structure with nonzero $c(\xi)$. This HF-driven framework provides a broad mechanism ensuring tightness for all non-positive surgeries and advances understanding of tight contact structures on hyperbolic manifolds.

Abstract

For any knot $K$ in $S^3$ and any positive rational $r$, we show that smooth $(-r)$-surgery on $K$ always admits a tight contact structure. More specifically, the tightness is detected by the non-vanishing Heegaard Floer contact invariant.

Figures (2)

• Figure 1: $n$ in the second diagram indicates $n$ zigzags (i.e. $n$ negative-stabilizations). We call the Legendrian knot, along which the $\frac{n+1}{n}$-surgery is performed in the first diagram, the standard Legendrian meridian of the knot which $e_1$ belongs to.
• Figure 2: The Legendrian surgery on $L_i^{n}$ is obtained by contact $(+\frac{n+1}{n})$-surgery (a sequence of $(+2)$-contact surgeries) on the standard Legendrian meridian of the Legendrian surgery on $L_i$.

Theorems & Definitions (39)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Definition 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Corollary 1.9
• Remark 1.10