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Traczyk used the first coefficient of the skein (Homfly) polynomial to find powerful criteria for r periodic knots. The criteria were extended by Yokota to (r 1)/2 first coefficients of the skein polynomial. We give here a short, elementary proof of the Traczyk-Yokota criteria. The main tool is the Jaeger composition product, the same product which is a base for Turaev's Hopf algebra structure of links in a handlebody. An oriented link L in S3 is said to be r-period (r an odd, prime number) if it has a diagram which is invariant under the rotation, Ao, of R2 by a 2 angle. Let lk(L, y) denote the linking number of L with the rotation axis, y . The skein polynomial of oriented links, PL E Z[V?1, Z1 ], is characterized by the properties: PT, = (V )n-1, where Tn is the trivial link of n components, and V-IPL+ VPL= ZPLO, where L+, L_, and Lo make up the standard skein triple. If K is a knot, then PK = Ei=0 P2i (K) z2i, where P2&(K) E Z[v?2] . Traczyk [4] and Yokota [5] proved the following property of the skein polynomial of periodic knots: Theorem 1. Let K be an r-periodic knot and k = lk(L, y). Then: (a) (Traczyk) If Po(K) = E a2iv2i then a2i a2i+2 mod r except possibly when 2i+1_kmodr. (b) (Yokota) P2&(K) _ b2iPo(K) mod r for 2i < r 1, where numbers b2i depend only on r and on k mod r. Notice that because K is a knot, therefore k is coprime with r. The crucial part of the proof, by Traczyk and Yokota, is an involved analysis of the skein polynomial of torus knots (using the Jones formula [2]). In our proof we do not need a reduction to torus knots at all and instead use the Jaeger composition product formula [1] (we should stress that Jaeger proved his formula by elementary considerations involving Reidemeister moves). It is convenient to work with the following regular isotopy variant of the Received by the editors August 4, 1993. 1991 Mathematics Subject Classification. Primary 57M25.

An elementary proof of the Traczyk-Yokota criteria for periodic knots