${\mathrm{S}\mathrm{L}}_N$ Quantum-Torus Summands and Visible Nielsen Numbers

Let $M_{\gamma }=T^2{\times }_{\gamma }{S}^1$, where $\gamma \in {\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$ is hyperbolic. For every $N\ge 2$, we compute the empty-skein, or quantum-torus, direct summand in Kinnear’s decomposition of the ${\mathrm{S}\mathrm{L}}_N$-skein module of $M_{\gamma }$, thereby answering his centralizer question for this summand in the hyperbolic case. Its dimension is expressed in terms of the periodic Nielsen numbers $N_k=\mathrm{\mid }\det (I-{\gamma }^k)\mathrm{\mid }$ and the visible Nielsen numbers $V_{e{\mathbb{Z}}^2}(f_{\gamma })$ carried by torsion in the Weyl coinvariant lattices. Only moduli $e\mid N$ occur, so the rank-$N$ summand is determined by $N_1,\dots ,N_N$ together with the visible Nielsen numbers at the divisors of $N$. On the ${\mathrm{G}\mathrm{L}}_N$ permutation lattice these coinvariants are torsion-free, so no such correction occurs; the observer corrections arise precisely upon passage to the ${\mathrm{S}\mathrm{L}}_N$ character lattice. For $N=3$, we obtain an explicit formula with the single correction $V_{3{\mathbb{Z}}^2}(f_{\gamma })$, and construct infinitely many pairs of non-homeomorphic hyperbolic torus bundles whose ${\mathrm{G}\mathrm{L}}_N$-skein-module dimensions agree for every $N$, while their ${\mathrm{S}\mathrm{L}}_3$ 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 ${\mathrm{S}\mathrm{L}}_N$-skein module.