Conway algebra

An abstract algebra which yields an invariant of links in (cf. also Link).

The concept is related to the entropic right quasi-group (cf. also Quasi-group). 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 four-element Conway algebra, which leads to the link invariant distinguishing the left-handed and right-handed trefoil knots (cf. also Torus knot) is described below:

The operations and are given by the following tables:'

<tbody> </tbody> 1 2 3 4 1 2 1 4 3 2 3 4 1 2 3 1 2 3 4 4 4 3 2 1

'

<tbody> </tbody> 1 2 3 4 1 3 1 2 4 2 1 3 4 2 3 2 4 3 1 4 4 2 1 3

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