By Shiing-Shen Chern

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**Example text**

In other words, while passing from a circle to itself one should pass even number of twistings. The latter follows from the fact that for a polygon with an odd number of sides one cannot choose the orientation of sides in such a way that any two adjacent sides have opposite orientations. 4. The parity of the number of Seifert surfaces for the diagram of a k-component link with n crossings coincides with the parity of n - k. Proof. Let L be a diagram of a k-component link with n vertices. Consider a Seifert surface S(L).

Thus we have defined the homomorphism ยข: r(ADIR D ) ---+ fK. In order to define the inverse homomorphism 1/J : f K ---+ r(A D I R D ), let us fix s E f K. Then, the path representing s is constructed in such a way that the projection of the path intersects D transversely and contains no diagram crossing. Denote by an, an-l, ... , al those arcs of D going over the path s. Denote by ao the arc corresponding to the end of s. Now, for each s E fK, let us assign the element (( ... (aoElal)E2 ... an-l)Ena n of the quandle r(ADIR D ), where Ei means / if s goes under ai from the left to the right, or 0, otherwise, see Fig.

2004 by CRC Press LLC Thus, Dehn's theorem reduces the trivial link recognition problem to the free group recognition problem (for some class of groups). In the general case, the free group recognition problem is undecidable. For more details see [Bir] and [BZ]. Dehn's theorem follows from the following lemma. 2. Let M be a 3-manifold with boundary and let'Y be a closed curve on its boundary 8M. Then if there exists an immersed 2-disc D ----+ M, such that 3D ~ 'Y then there exists an embedded disc D ' c M with the same boundary 3D'~'Y.