Knots are commonly represented and manipulated via diagrams, which are decorated planar graphs. When such a knot diagram has low treewidth, parameterized graph algorithms can be leveraged to ensure the fast computation of many invariants and properties of the knot. It was recently proved that there exist knots which do not admit any diagram of low treewidth, and the proof relied on intricate low-dimensional topology techniques. In this work, we initiate a thorough investigation of tree decompositions of knot diagrams (or more generally, diagrams of spatial graphs) using ideas from structural graph theory. We define an obstruction on spatial embeddings that forbids low tree width diagrams, and we prove that it is optimal with respect to a related width invariant. We then show the existence of this obstruction for knots of high representativity, which include for example torus knots, providing a new and self-contained proof that those do not admit diagrams of low treewidth. This last step is inspired by a result of Pardon on knot distortion.

翻译：纽结通常通过图论中的装饰平面图——即纽结图表——来表示和操作。当这类纽结图表具有低树宽时，可借助参数化图算法高效计算纽结的众多不变量和性质。近期有学者证明存在某些纽结无法用任何低树宽图表表示，其证明依赖于复杂的低维拓扑技术。本研究首次运用结构图论思想，系统探究纽结图表（更广义地，空间图图表）的树分解问题。我们定义了空间嵌入中对低树宽图表的障碍条件，并证明其相对于相关宽度不变量具有最优性。进而证明高表示度纽结（包括环面纽结等）存在该障碍，从而以自包含方式证明这些纽结不存在低树宽图表。最后一步的证明思路受Pardon关于纽结扭曲性研究的启发。