Instanton dimensions of knot surgeries over arbitrary fields
Zhenkun Li, Fan Ye
TL;DR
The paper proves a universal dimension formula for framed instanton homology of knot surgeries over arbitrary fields, introducing concordance invariants $\nu^{\sharp}_{\mathbb{K}}(K)$ and $r_{\mathbb{K}}(K)$ that govern the growth of dimensions in $p/q$-sloped surgeries via $\dim I^{\sharp}(S^3_{p/q}(K);\mathbb{K}) = q\,r_{\mathbb{K}}(K) + |p - q\,\nu^{\sharp}_{\mathbb{K}}(K)|$, with an exceptional case when $p/q$ equals $\nu^{\sharp}_{\mathbb{K}}(K)$ and the index is even. It extends Baldwin--Sivek’s complex-field result to general fields, organizes the surgery theory through bundle-sets and a distance-two surgery triangle, and analyzes implications for $SU(2)$-representations, showing strong constraints on when surgeries can be $SU(2)$-abelian. The work establishes a robust framework for rational surgeries, proves a generalized distance-two triangle, and applies these tools to deduce properties of instanton L-space knots, genus-one knots, and $SU(2)$-averse knots. Overall, the paper deepens the connection between instanton Floer theory, representation varieties, and knot-surgery topology across diverse coefficient rings, with concrete constraints on representation-theoretic behaviors of surgered manifolds.
Abstract
Suppose $K \subset S^3$ is a knot and suppose $p$ and $q$ are co-prime integers with $q\ge 1$. For any field $\mathbb{K}$, we establish a dimension formula for the framed instanton homology of knot surgeries: $$ \dim I^\sharp(S^3_{p/q}(K); \mathbb{K}) = q \cdot r_{\mathbb{K}}(K) + |p - q \cdot ν^\sharp_{\mathbb{K}}(K)| $$ for certain integers $r_{\mathbb{K}}(K)$ and $ν^\sharp_{\mathbb{K}}(K)$, except possibly when $p/q = ν^\sharp_{\mathbb{K}}(K)$ and $ν^\sharp_{\mathbb{K}}(K)$ is even. This formula generalizes the result of Baldwin--Sivek from the case $\mathbb{K} = \mathbb{C}$ to arbitrary fields. Based on the result for $\mathbb{K} = \mathbb{Z}/2$, we obtain that $S^3_{p/q}(K)$ is not $SU(2)$-abelian for any knot $K$ other than the unknot and the right-handed trefoil whenever $p/q \in [0,6)$ and $p \in \{ a^e, 2a^e \}$ for some prime number $a$ and natural number $e$, thereby extending existing results for $p/q \in [0,5]$ and $p = a^e$. A byproduct of the techniques developed in this paper is that we generalize the distance-two surgery exact triangle by Culler--Daemi--Xie and Daemi--Miller-Eismeier--Lidman from $\mathbb{Z}/2$ coefficients to any coefficient ring.