Tadayuki Watanabe

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.

Knotted trivalent graphs and construction of the LMO invariant from triangulations

We give a Turaev-Viro type construction for the LMO invariant. More precisely, we construct an invariant of closed oriented 3-manifolds from data of their spines or their simplicial decompositions and the values of Kontsevich invariant of the unknotted tetrahedron and the Hopf link by using Bar-Natan and Thurston’s operations.