The Kontsevich invariant is a knot invariant which is universal among all quantum invariants of knots and all Vassiliev invariants of knots. The Kontsevich invariants is a very strong invariant, and is expected to classify all knots. However, the value of the Kontsevich invariant is presented by an in finite sum of Jacobi diagrams, and in general, it is so difficult to determine all terms of its value at the same time. On the other hand, the Kontsevich invariant has a special expansion, which is called the loop expansion. This expansion gives a strong restriction of the image of the Kontsevich invariant, and can be one approach to determine the the value of the Kontsevich invariant. In this talk, I would explain the Kontsevich invariant, its loop expansion, and the some results knownabout them.
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