Traceless $\mathrm{SU}(2)$ characters and $\mathbb{Z}/4$ instanton gradings for two-bridge and $(3,n)$-torus knots

Abstract

For the $(3,n)$-torus knots we determine the $\mathbb{Z}/4$ gradings of the reduced singular instanton chain complex through the double branched cover $\Sigma(2,3,n)$ rather than through an index computation: each irreducible flat connection on the cover has two traceless knot lifts of equal grading, so Daemi-Scaduto's theorem for the irreducible torus-knot complex transports to a grading split. For $n$ odd this recovers the chain-rank distribution $(1+a,a,a,a)$, $a=-\sigma/4$, conjectured by Poudel-Saveliev and established for all torus knots by Daemi-Scaduto. Comparing that rank vector with $\operatorname{rank}I^{\natural}=\|\Delta\|_1$ then determines the total rank of the framed differential in every residue class of $n$ modulo $6$: it vanishes for $n\equiv1,2$ and equals one for $n\equiv4,5$. The comparison uses no index theory, which is what carries it into the even classes, where $\Sigma(2,3,n)$ is not a homology sphere; $T(3,8)$ is a non-alternating knot with vanishing framed differential. Along the way we prove that every irreducible traceless $\mathrm{SU}(2)$ character of a two-bridge knot is binary-dihedral, with the traceless Riley polynomial in closed form, and that for the $(3,n)$-torus knots exactly $(\det-1)/2$ characters are dihedral, so for $n$ odd none is; and we identify the first nonzero pillowcase differential, for $8_{19}=T(3,4)$, as a corner figure-eight bigon absent on two-bridge knots.