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

Abstract

For every torus knot, the rank of the differential of the Daemi-Scaduto S-complex of singular instanton homology is determined by the signature and the Alexander polynomial; this follows from theorems of Li-Ye and Daemi-Scaduto. For the torus knots $T(3,n)$ the differential has rank zero or one according to $n$ modulo $6$, and its rank equals Daemi and Scaduto's Frøyshov-type invariant; they proved this in two of the four residue classes and for $n=4,5$, and Hedden, Herald and Kirk computed it for $n\le 38$. Following Lewallen, Klassen and Hedden-Herald-Kirk, we describe the generators of the instanton complexes and their $\mathbb{Z}/4$ gradings for two-bridge knots, whose irreducible traceless characters are all binary dihedral, and for $T(3,n)$, where at most one is. We also give a proof through Khovanov homology, using Schütz's computation for three-braids, of Li and Ye's computation of the rank of the instanton homology of $T(3,n)$, and compare the differential of the knot $8_{19}$ with the single pillowcase bigon of Hedden, Herald and Kirk.

arXiv comment15 pages, 1 figure