Planar diagrams for local invariants of graphs in surfaces

In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These virtual graphs are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, the [Formula: see text]-polynomial, and formulate the [Formula: see text] Penrose polynomial for non-cubic graphs, giving contraction–deletion relations. The [Formula: see text]-polynomial is used to define an extension of the Yamada polynomial to virtual spatial graphs, and with it we obtain a sufficient condition for non-classicality of virtual spatial graphs. We conjecture the existence of local relations for the [Formula: see text]-polynomial at squares of integers.

Walker's Cancellation Theorem

Walker's cancellation theorem says that if B + Z isisomorphic to C + Z in the category of abeliangroups, then B is isomorphic to C. We construct an example ina diagram category of abelian groups where the theorem fails. As aconsequence, the original theorem does not have a constructiveproof. In fact, in our example B and C are subgroups ofZ<sup>2</sup>. Both of these results contrast with a group whoseendomorphism ring has stable range one, which allows aconstructive proof of cancellation and also a proof in any diagramcategory.