2026/07/28 by Bernd J. Wuebben
Mathematics · #math.GT #math.SG #msc:53D40 #msc:57K10 #msc:57K31 #msc:57R58 #msc:58J30
10 pages, 1 figure
arxiv created 2026/07/31 · arxiv updated 2026/08/03
We assemble, and where possible independently verify, the representation-theoretic data underlying the pillowcase (symplectic) side of the Atiyah-Floer conjecture for knots, for two-bridge knots and (3,n)-torus knots. For a two-bridge knot b(p,q) we give a short self-contained proof that every irreducible traceless SU(2) representation is binary-dihedral; these are the (p-1)/2 dihedral characters at meridian angles cos(2πk/p), independent of q, and the traceless Riley polynomial is the explicit product ϕp(u)=∏k (u+4sin2(πk/p)), monic of degree (p-1)/2 with constant term det K. This gives a transparent account of the Hedden-Herald-Kirk theorem that pillowcase homology equals reduced singular instanton knot homology I^\natural on this family, and of why the figure-eight bubbling and bounding cochains obstructing the general conjecture are structurally inert there. For the (3,n)-torus knots we compute the full traceless character variety and prove a dichotomy: exactly (det-1)/2 characters are dihedral, so for n odd every irreducible traceless character is non-dihedral. Passing to the double branched cover Σ(2,3,n), we self-compute the ℤ/4 spectral-flow gradings of the generators from the Fintushel-Stern index and the equivariant ρ-invariant, calibrated against the Poudel-Saveliev and Anvari computations; for n odd the gradings split evenly between 1 and 3, giving the chain complex IC^\natural(T(3,n))=(1+a,a,a,a), a=-σ/4. The homology equals this for n≡ 1\pmod 6 (differential zero) but is smaller by 2 for n≡ 5, where it is nonzero: rank I^\natural=∑i|ΔT(3,n)| throughout, and T(3,5)=P(-2,3,5)=10124 has rank 7, not 9. We reproduce the first nonzero pillowcase differential, for 819=T(3,4), and identify it as the corner figure-eight bigon absent on two-bridge knots.