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）进行表示与操作。当此类纽结图具有低树宽时，可利用参数化图算法快速计算纽结的许多不变量与性质。近期研究表明，存在不承认任何低树宽图的纽结，其证明依赖于复杂的低维拓扑技术。本文中，我们利用结构图论的思想，系统研究纽结图（或更一般地，空间图）的树分解。我们定义了一个空间嵌入中的障碍，该障碍禁止低树宽图的存在，并证明其相对于相关宽度不变量达到最优。随后，我们证明了高代表纽结（例如环面纽结）存在该障碍，从而提供了一种新的自包含证明，表明此类纽结不承认低树宽图。最后一步的灵感来源于Pardon关于纽结扭曲性的结果。