The S-complex differential of torus knots

For a torus knot \(K=T(p,q)\) the traceless \(\mathrm{SU}(2)\) 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 \(I^{\natural}(K)\) has rank \(\|\Delta_K\|_1\), the sum of the absolute values of the coefficients of the Alexander polynomial; so the differential has rank \(R(p,q)=\frac12(1+|\sigma|-\|\Delta\|_1)\). 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 \(q\equiv2\) or \(q\equiv p+2(-1)^{(p-1)/2}\pmod{2p}\); 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 \(I^{\natural}(T(p,q);\mathbb{Q})\) depends only on \(\sigma\) and \(\|\Delta\|_1\), 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 \(2\).