The S-complex differential of torus knots

Abstract

For a torus knot $K=T(p,q)$ the traceless $\mathrm{S}\mathrm{U}(2)$ character variety is finite and nondegenerate, so Daemi and Scaduto’s S-complex for singular instanton homology can be built on it. The resulting complex has rank $1+|\sigma (K)|$, where $\sigma$ is the signature, and by theorems of Li and Ye and of Kronheimer and Mrowka its homology $I^{♮}(K)$ has rank $\Vert {\Delta }_K{\Vert }_1$, the sum of the absolute values of the coefficients of the Alexander polynomial; so the differential has rank $R(p,q)=\frac{1}{2}(1+|\sigma |-\Vert \Delta {\Vert }_1)$. We express $R(p,q)$ through floor sums and show that, for coprime $2\le p<q$, $R(p,q)\ge \lfloor (p-2b{)}^2/4\rfloor$, where $b\in \{1,\dots ,p-1\}$ is the inverse of $q$ modulo $p$, and that the differential vanishes if and only if $p=2$, or $p$ is odd and $q\equiv 2$ or $q\equiv p+2(-1{)}^{(p-1)/2}(\mathrm{mod}2p)$; the ‘if’ direction is due to Daemi and Scaduto. By a result of Daemi and Scaduto, the component of the differential to the reducible generator is nonzero over $\mathbb{Q}$ exactly when their invariant $h$ is positive. We show that the components from degree $3$ to degree $2$ and from degree $1$ to degree $0$ have ranks $\lfloor R/2\rfloor$ and $\lceil R/2\rceil$, independently of the choices in the construction. In particular the $\mathbb{Z}/4$-graded group $I^{♮}(T(p,q);\mathbb{Q})$ depends only on $\sigma$ and $\Vert \Delta {\Vert }_1$, and we give it explicitly. In each small-perturbation pillowcase computation of Hedden, Herald and Kirk, a bigon ends at the reducible generator exactly when $h>0$. We formulate a chain-level conjecture identifying the two complexes over ${\mathbb{F}}_2$; it predicts this pattern whenever the map to the reducible is nonzero modulo $2$.