On a move reducing the genus of a knot diagram

For a knot diagram we introduce an operation which does not increase the genus of the diagram and does not change its representing knot type. We also describe a condition for this operation to certainly decrease the genus. The proof involves the study of a relation between the genus of a virtual knot diagram and the genus of a knotoid diagram, the former of which has been introduced by Stoimenow, Tchernov and Vdovina, and the latter by Turaev recently. Our operation has a simple interpretation in terms of Gauss codes and hence can easily be computer-implemented.

1-loop graphs and configuration space integral for embedding spaces

We will construct differential forms on the embedding spaces Emb(R^j,R^n) for n-j>=2 using configuration space integral associated with 1-loop graphs, and show that some linear combinations of these forms are closed in some dimensions. There are other dimensions in which we can show the closedness if we replace Emb(R^j,R^n) by fEmb(R^j,R^n), the homotopy fiber of the inclusion Emb(R^j,R^n) -> Imm(R^j,R^n). We also show that the closed forms obtained give rise to nontrivial cohomology classes, evaluating them on some cycles of Emb(R^j,R^n) and fEmb(R^j,R^n). In particular we obtain nontrivial cohomology classes (for example, in H^3(Emb(R^2,R^5))) of higher degrees than those of the first nonvanishing homotopy groups.