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

Abstract

Let $M_\gamma=T^2\times_\gamma S^1$, where $\gamma\in\mathrm{SL}_2(\mathbb{Z})$ is hyperbolic. For every $N\geq2$, we compute the empty-skein, or quantum-torus, direct summand in Kinnear's decomposition of the $\mathrm{SL}_N$-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 $N_k=|\det(I-\gamma^k)|$ 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 $N_1,\ldots,N_N$ together with the visible Nielsen numbers at the divisors of $N$. On the $\mathrm{GL}_N$ permutation lattice these coinvariants are torsion-free, so no such correction occurs; the observer corrections arise precisely upon passage to the $\mathrm{SL}_N$ 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 $\mathrm{GL}_N$-skein-module dimensions agree for every $N$, while their $\mathrm{SL}_3$ 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 $\mathrm{SL}_N$-skein module.