Knot surgery formulae for instanton Floer homology I: the main theorem

TL;DR

This paper establishes a new integral surgery formula for framed instanton homology of knots in 3-manifolds, using sutured instanton homology and derived category techniques, with implications for exact triangles and further applications.

Contribution

It introduces a novel proof of the integral surgery formula for instanton homology, employing sutured instanton homology and octahedral diagrams, expanding the toolkit for knot surgery analysis.

Findings

Derived new exact triangles for sutured instanton homology.

Established an exact triangle relating different surgery manifolds.

Connected surgery cobordism maps to bypass maps.

Abstract

We prove an integral surgery formula for framed instanton homology $I^\sharp(Y_m(K))$ for any knot $K$ in a $3$-manifold $Y$ with $[K]=0\in H_1(Y;\mathbb{Q})$ and $m\neq 0$. Though the statement is similar to Ozsv\'ath-Szab\'o'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 a corollary, we obtain an exact triangle between $I^\sharp(Y_m(K))$, $I^\sharp(Y_{m+k}(K))$ and $k$ copies of $I^\sharp(Y)$ 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.