The goal of this paper is to construct distinct trisections of the same genus on a fixed 4-manifold. For every \(k \ge 2\), we exhibit infinitely many manifolds with \(2^{k}-1\) non-diffeomorphic (3k, k)-trisections. Here, the manifolds are spun Seifert fiber spaces and the trisections come from Meier’s spun trisections. The technique used to distinguish the trisections parallels a common setup for distinguishing Heegaard splittings. In particular, we show that the Nielsen classes of the generators of the fundamental group, obtained from spines of the 4-dimensional 1-handlebodies of the trisection, are isotopy invariants of the trisection. If we additionally consider the action of the automorphism group on the Nielsen classes we obtain diffeomorphism invariants. Once the invariance is established, we analyze the Nielsen classes of the spun trisections in terms of the Nielsen classes of the original Heegaard splitting, and leverage work of Lustig, Moriah, and Rosenberger on Nielsen classes of Fuchsian groups in order to distinguish the trisections.

本文的目的是在固定的4流形上构造同一属的不同三等分。对于每个\（k \ ge 2 \），我们表现出无穷多的流形具有\（2 ^ {k} -1 \）非微分（3 k，  k）-三等分。在这里，歧管是纺制的Seifert纤维空间，三等分来自Meier的纺制三等分。用于区分三等分的技术与区分Heegaard分裂的通用设置相似。尤其是，我们表明，从三叉戟的4维1 le体的刺中获得的基群生成器的Nielsen类是三叉戟的同位素不变量。如果我们还考虑自同构群对Nielsen类的作用，我们将获得不等构不变量。一旦建立了不变性，我们就根据原始Heegaard分裂的Nielsen类分析旋转三等分的Nielsen类，并利用Lustig，Moriah，