Polyak-Viro type formulas for high-(co)dimensional smooth knots and links

I will talk about the problem of finding a high-dimensional analogue to Polyak-Viro type formulas given in the classical case of 1-dimensional knots in R^3. We obtained such formulas for invariants of 2- and 3-component links of dimension (2m-1) in R^{3m} for m>1. At the end, I will give a conjectural formula for the Haefliger invariant for embeddings of S^3 in R^6.