Conway algebra
An abstract algebra which yields an invariant of links in (cf. also Link).
The concept is related to the entropic right quasigroup (cf. also Quasigroup). A Conway algebra consists of a sequence of argument operations (constants) and two argument operations and , which satisfy the following conditions:
Initial conditions:
C1) ;
C2) .
Transposition properties:
C3) ;
C4) ;
C5) .
Inverse operation properties:
C6) ;
C7) . The main link invariant yielded by a Conway algebra is the Jones–Conway polynomial, [a2], [a5], [a4].
A nice example of a fourelement Conway algebra, which leads to the link invariant distinguishing the lefthanded and righthanded trefoil knots (cf. also Torus knot) is described below:
The operations and are given by the following tables:'
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'
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If one allows partial Conway algebras, one also gets the Murasugi signature and Tristram–Levine signature of links [a3]. The skein calculus (cf. also Skein module), developed by J.H. Conway, leads to the universal partial Conway algebra.
Invariants of links, , yielded by (partial) Conway algebras have the properties that for the Conway skein triple , and :
References
[a1]  J.H. Conway, "An enumeration of knots and links" J. Leech (ed.) , Computational Problems in Abstract Algebra , Pergamon (1969) pp. 329–358 
[a2]  J.H. Przytycki, P. Traczyk, "Invariants of links of Conway type" Kobe J. Math. , 4 (1987) pp. 115–139 
[a3]  J.H. Przytycki, P. Traczyk, "Conway algebras and skein equivalence of links" Proc. Amer. Math. Soc. , 100 : 4 (1987) pp. 744–748 
[a4]  A.S. Sikora, "On Conway algebras and the Homflypt polynomial" J. Knot Th. Ramifications , 6 : 6 (1997) pp. 879–893 
[a5]  J.D. Smith, "Skein polynomials and entropic right quasigroups Universal algebra, quasigroups and related systems (Jadwisin 1989)" Demonstratio Math. , 24 : 1–2 (1991) pp. 241–246 
Conway algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conway_algebra&oldid=18395