2026/08/03 by Ahmet Selman Kaya
Mathematics · #math.QA #msc:57K31 #msc:55M20 #msc:57R56 #msc:16E40 #msc:20C30
arxiv created 2026/08/03 · arxiv updated 2026/08/05
Let Mγ=T2×γS1, where γ\inSL2(ℤ) is hyperbolic. For every N≥2, we compute the empty-skein, or quantum-torus, direct summand in Kinnear's decomposition of the SLN-skein module of Mγ, thereby answering his centralizer question for this summand in the hyperbolic case. Its dimension is expressed in terms of the periodic Nielsen numbers Nk=|det(I-γk)| and the visible Nielsen numbers Veℤ2(fγ) carried by torsion in the Weyl coinvariant lattices. Only moduli e| N occur, so the rank-N summand is determined by N1,…,NN together with the visible Nielsen numbers at the divisors of N. On the GLN permutation lattice these coinvariants are torsion-free, so no such correction occurs; the observer corrections arise precisely upon passage to the SLN character lattice. For N=3, we obtain an explicit formula with the single correction V3ℤ2(fγ), and construct infinitely many pairs of non-homeomorphic hyperbolic torus bundles whose GLN-skein-module dimensions agree for every N, while their SL3 quantum-torus summands differ in dimension by six. These pairs also have identical periodic Nielsen data and finite-cover visibility profiles at every iterate. We do not compute the additional endomorphism-algebra summands of the full SLN-skein module.