Vassiliev's knot invariants can be computed in different ways but many of them as Kontsevich integral are very difficult. We consider more visual diagram formulas of the type Polyak-Viro and give new diagram formula for the two basic Vassiliev invariant of degree 4.

The possible mechanisms for anomalous decay Z+Z + l~y are discussed.

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Abstract. We describe the algebra of finite order invariants on the set of all (n, 2)-torus knots. * This paper is an extended exposition of the talk [Ty] given by the first author. The authors thank S. Duzhin and A. Sossinsky for interest to this work and S. Chmutov for useful discussions.Consider the Q-algebra V of Vassiliev finite order knot invariants, see for example [B, CDL]. The algebra is filtered,The subspace V 0 is of dimension 1 and consists of invariants taking the same value on all knots. It is known that V 1 = V 0 , dim V 2 /V 1 = dim V 3 /V 2 = 1. The generator in V 2 /V 1 is given by the knot invariant x of order 2 which takes value 0 on the trivial knot and value 8 on the trefold. The generator in V 3 /V 2 is given by the knot invariant y of order 3 which takes value 0 on the trivial knot, takes value 24 on the trefold, and takes value −24 on its mirror image. Those conditions determine x and y uniquely, see for example [L].It is known that the space V k has finite dimension fast growing with k, see for example [CD, D, Z].By definition the algebra V is an algebra of certain special functions on the set K of all knots in R 3 considered up to isotopy.