Writhe invariants of 3-regular spatial graphs

Abstract

We give a necessary condition for two diagrams of $3$-regular spatial graphs with the same underlying abstract graph $G$ to represent isotopic spatial graphs. The test works by reading off the writhes of the knot diagrams coming from a collection of cycles in $G$ in each diagram, and checking whether the writhe tuples differ by an element in the image of a certain map of $\mathbb{Z}$-modules determined by $G$. We exemplify by using our result to distinguish, for each $n \ge 3$, all elements in a certain infinite family of embeddings of the M\"obius ladder $\mathrm{M}_n$ into $\mathbb{R}^3$ . We also connect these writhe tuples to a classical invariant of spatial graphs due to Wu and Taniyama.