3-manifolds are commonly represented as triangulations, consisting of abstract tetrahedra whose triangular faces are identified in pairs. The combinatorial sparsity of a triangulation, as measured by the treewidth of its dual graph, plays a fundamental role in the design of parameterized algorithms. In this work, we investigate algorithmic procedures that transform or modify a given triangulation while controlling specific sparsity parameters. First, we revisit a standard, linear-time algorithm that converts a given triangulation into a Heegaard diagram of the underlying 3-manifold, showing that the construction preserves treewidth. We apply this construction to exhibit a fixed-parameter tractable framework for computing Kuperberg's quantum invariants of 3-manifolds. Second, we present a quasi-linear-time algorithm that retriangulates a given triangulation into one with maximum edge valence of at most nine, while only moderately increasing the treewidth of the dual graph. Combining these two algorithms yields a quasi-linear-time algorithm that produces, from a given triangulation, a Heegaard diagram in which every attaching curve intersects at most nine others.

翻译：三维流形通常表示为三角剖分，即由抽象四面体构成，其三角形面成对粘合。三角剖分的组合稀疏性（以其对偶图的树宽度量）在参数化算法设计中起着基础性作用。本文研究在控制特定稀疏性参数的同时，对给定三角剖分进行变换或修改的算法过程。首先，我们重新审视一个标准的线性时间算法，该算法将给定三角剖分转换为底层三维流形的Heegaard图，并证明该构造保持树宽不变。我们应用此构造，为计算Kuperberg的三维流形量子不变量展示了一个固定参数可处理框架。其次，我们提出一个拟线性时间算法，该算法将给定三角剖分重新三角化为最大边价至多为九的三角剖分，同时仅适度增加其对偶图的树宽。结合这两个算法，我们得到一个拟线性时间算法，该算法从给定三角剖分生成一个Heegaard图，其中每条附着曲线至多与九条其他曲线相交。