Knot surgery formulae for instanton Floer homology, I: The main theorem

Li, Zhenkun; Ye, Fan

doi:10.2140/gt.2025.29.2269

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

Author index

To appear

Other MSP journals

Knot surgery formulae for instanton Floer homology, I: The main theorem

Zhenkun Li and Fan Ye

Geometry & Topology 29 (2025) 2269–2342

DOI: 10.2140/gt.2025.29.2269

Abstract

We prove an integral surgery formula for framed instanton homology $I^{♯} (Y_{m} (K))$ for any knot $K$ in a $3$ -manifold $Y$ with $[K] = 0 \in H_{1} (Y; ℚ)$ and $m \neq 0$ . Although the statement is similar to Ozsváth–Szabó’s integral surgery formula for Heegaard Floer homology, the proof is new and based on sutured instanton homology $SHI$ and the octahedral lemma in the derived category. As byproducts, we obtain a formula computing instanton knot homology of the dual knot analogous to Eftekhary’s and Hedden–Levine’s work, and also an exact triangle between $I^{♯} (Y_{m} (K))$ , $I^{♯} (Y_{m + k} (K))$ and $k$ copies of $I^{♯} (Y)$ for any $m \neq 0$ and large $k$ . In the proof of the formula, we discover many new exact triangles for sutured instanton homology and relate some surgery cobordism map to the sum of bypass maps, which are of independent interest. In a companion paper, we derive many applications and computations based on the integral surgery formula.

Keywords

instanton, Floer homology, Dehn surgery, mapping cone formula

Mathematical Subject Classification

Primary: 57K10, 57K14, 57K16, 57K18, 57K31

References

Publication

Received: 27 September 2022

Revised: 3 August 2024

Accepted: 14 September 2024

Published: 14 August 2025

Proposed: András I Stipsicz

Seconded: Ciprian Manolescu, Peter Ozsváth

Authors

Zhenkun Li
Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

Fan Ye
Department of Mathematics Harvard University Cambridge, MA United States

Open Access made possible by participating institutions via Subscribe to Open.

Volume 29, issue 5 (2025)