Traceless SU(2) characters and 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≤38. Following Lewallen, Klassen and Hedden-Herald-Kirk, we describe the generators of the instanton complexes and their 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 819 with the single pillowcase bigon of Hedden, Herald and Kirk.
Related
- Indecomposable characters of the wreath product of a torus by the infinite block-diagonal symmetric group
- Compatible gradings on dg models and cluster equivalence
- Algebraicity, Hodge Gradings, and the Geometry of Period Images
- Group gradings, graded identities and central polynomials of algebras with almost polynomial δ-growth
- Okubo algebras and Exceptional Periodicity: tetracode gradings, root-lattice obstructions and Clifford cones
BibTeX
@misc{wuebben2026traceless,
title = {{Traceless $\mathrmSU(2)$ characters and $\mathbbZ/4$ instanton gradings for two-bridge and $(3,n)$-torus knots}},
author = {Bernd Johannes Wuebben},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.26095},
eprint = {2607.26095},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.26095},
}
RIS
TY - UNPB
TI - Traceless SU(2) characters and ℤ/4 instanton gradings for two-bridge and (3,n)-torus knots
AU - Bernd Johannes Wuebben
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2607.26095
UR - https://arxiv.org/abs/2607.26095
AB - 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≤38. Following Lewallen, Klassen and Hedden-Herald-Kirk, we describe the generators of the instanton complexes and their 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.
ER -