Instanton dimensions of knot surgeries over arbitrary fields

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: $$\dim I^{\mathrm{♯}}(S_{p\mathrm{/}q}^3(K);\mathbb{K})=q\cdot r_{\mathbb{K}}(K)+\mathrm{\mid }p-q\cdot {\nu }_{\mathbb{K}}^{\mathrm{♯}}(K)\mathrm{\mid }$$ for certain integers $r_{\mathbb{K}}(K)$ and ${\nu }_{\mathbb{K}}^{\mathrm{♯}}(K)$, except possibly when $p\mathrm{/}q={\nu }_{\mathbb{K}}^{\mathrm{♯}}(K)$ and ${\nu }_{\mathbb{K}}^{\mathrm{♯}}(K)$ 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 $\mathbb{K}=\mathbb{Z}\mathrm{/}2$, we obtain that $S_{p\mathrm{/}q}^3(K)$ is not $SU(2)$-abelian for any knot $K$ other than the unknot and the right-handed trefoil whenever $p\mathrm{/}q\in [0,6)$ and $p\in \{a^e,2a^e\}$ for some prime number $a$ and natural number $e$, thereby extending existing results for $p\mathrm{/}q\in [0,5]$ 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 $\mathbb{Z}\mathrm{/}2$ coefficients to any coefficient ring.