Instanton dimensions of knot surgeries over arbitrary fields

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: $$\dim I^{♯}(S_{p/q}^3(K);\mathbb{K})=q\cdot r_{\mathbb{K}}(K)+|p-q\cdot \hat{I}{½}_{\mathbb{K}}^{♯}(K)|$$ for certain integers $r_{\mathbb{K}}(K)$ and $\hat{I}{½}_{\mathbb{K}}^{♯}(K)$, except possibly when $p/q=\hat{I}{½}_{\mathbb{K}}^{♯}(K)$ and $\hat{I}{½}_{\mathbb{K}}^{♯}(K)$ 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 $S_{p/q}^3(K)$ 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.

39 pages, no figures. Comments are welcome