Kontsevich invariant of surface knot Kontsevich defined knot invariants by iterated integrals in the first half of 1990s in
M. Kontsevich, Vassiliev's knot invariants, Adv. Sov. Math., 16(2) (1993) 137-150.
After that Le, Murakami and Ohtsuki defined powerful invariant for 3-manifold by using the method of Kontsevich integral.
Polyak defined integral type invariant for curves in https://arxiv.org/pdf/1108.4288.pdf
On the other hand, it seems that such kind of integral are not known for locally flatly embedding of closed surface in $R^4$ i.e.  surface knot.
Does anyone know surface knot invariants by configuration integrals ? 
(Except linking number)
 A: Here are some references:


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*Rossi, C. A. (2002). Invariants of higher-dimensional knots and
topological quantum field theories (Doctoral dissertation,
Universität Zürich, Institut für Mathematik).

*Cattaneo, A. S., & Rossi, C. A. (2005). Wilson surfaces and
higher dimensional knot invariants. Communications in mathematical
physics, 256(3), 513-537.

*Watanabe, T. (2007). Configuration space integral for long
n–knots and the Alexander polynomial. Algebraic & Geometric Topology,
7(1), 47-92.

*Sakai, K. (2010). Configuration space integrals for embedding
spaces and the Haefliger invariant. Journal of Knot Theory and Its
Ramifications, 19(12), 1597-1644.

*Sakai, K., & Watanabe, T. (2012). 1-loop graphs and
configuration space integral for embedding spaces. Mathematical
Proceedings of the Cambridge Philosophical Society, 152(3), 497-533.
These papers are continuation rather of the approach of Bott-Taubes (see Bott-Taubes and Volic), than the Kontsevich integral for knots. However, the two can be seen as outcomes of 'picking two different gauges' for perturbative expansions in the Chern-Simons gauge theory (see Labastida). To my knowledge it is still an open problem if the two universal knot invariants are the same.
There are also some results using different techniques (manifold calculus and operads):


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*Arone, G., & Turchin, V. (2014). On the rational homology of high-dimensional analogues of spaces of long knots. Geometry & Topology, 18(3), 1261-1322.

*Arone, G., & Turchin, V. (2015). Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots. In Annales de l'Institut Fourier, Volume 65 no. 1, pp. 1-62.
Here the formality of little-disks operads is used, and this is again very closely related to integrals over configuration spaces. To me its quite mysterios how exactly the two approaches are related.
