Knots are typically represented and manipulated through diagrams, which are decorated planar graphs. If the tree width of such a knot diagram is small, then parameterized graph algorithms can be exploited to quickly compute many knot invariants and properties. It was recently proved that there are knots that disallow low treewidth diagrams, and the proof relied on complex low-dimensional topological methods. In this work, we begin a thorough investigation of tree decompositions of knot diagrams (or, more generally, diagrams of spatial graphs) using ideas from structural graph theory. We define a spatial embedding obstacle that forbids diagrams with low tree widths and prove it to be optimal with respect to the relevant width invariants. We then show the existence of this obstacle for highly representative knots, including, for example, torus knots, and provide a new self-contained proof that they do not allow low-tree-width diagrams. This last step is inspired by Purdon’s results for knot distortion.

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