PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 100, Number 4, August 1987 CON WAY ALGEBRAS AND SKEIN EQUIVALENCE OF LINKS JÓZEF H. PRZYTYCKI AND PAWEL TRACZYK ABSTRACT. We consider a class of pairs of links which are not skein equivalent but have the same invariant in every Conway algebra. 1. Conway algebras. We will first recall the notion of Conway algebra as introduced in [PT]. DEFINITION 1.1. A Conway algebra is an algebra A with a sequence of 0- argument operations czi, a2,... and two 2-argument operations | and *, which sat- isfy the following conditions: Cl. an\an+i = an, C2. an * an+i = an (Cl and C2 are initial conditions properties), C3. (a\b)\(c\d) = (a]c)\(b\d), C4. (a\b) * (c\d) — (a * c)\(b * d) (C3, C4 and C5 are transposition properties), C5. (a * b) * (c * d) = (a * c) * (b * d), C6. (a\b)*b = a, C7. (a*b)\b = a. As shown in [PT] every Conway algebra yields an invariant of links which is constant on skein equivalence classes (skein invariant). It is uniquely determined by the following conditions: At„ = o-n (initial relations), Al+ — Al-\Al0 and Al„ = Al+ * Al0 (Conway relations). Here Tn denotes a trivial link of n components and L+, L_, and Lq are diagrams of oriented links identical except near one crossing point (see Figure 1.1). L+ L_ ¿0 Figure l.i 2. Skein equivalence and the signature. We recall that skein equivalence is the smallest equivalence relation ~s on the set of all oriented links in S3 such that: (i) if L is ambient isotopic to L', then L ~s L', Received by the editors December 10, 1985 and, in revised form, April 8, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M25. Key words and phrases. Skein equivalence, Jones polynomial, signature. The second author acknowledges support received during the preparation of this paper from the Matsumae International Foundation and Osaka University. ©1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page 744 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use