Conway algebra
An abstract algebra which yields an invariant of links in $\mathbf{R} ^ { 3 }$ (cf. also Link).
The concept is related to the entropic right quasigroup (cf. also Quasigroup). A Conway algebra consists of a sequence of $0$argument operations (constants) $a _ { 1 } , a _ { 2 } , \dots$ and two $2$argument operations $$ and $*$, which satisfy the following conditions:
Initial conditions:
C1) $a _ { n }  a _ {n + 1} = a _ { n }$;
C2) $a _ { n } * a _ { n + 1} = a _ { n }$.
Transposition properties:
C3) $( a  b )  ( c  d ) = ( a  c )  ( b  d )$;
C4) $( a  b ) * ( c  d ) = ( a * c )  ( b * d )$;
C5) $( a * b ) * ( c * d ) = ( a * c ) * ( b * d )$.
Inverse operation properties:
C6) $( a  b ) * b = a$;
C7) $( a * b )  b = a$. The main link invariant yielded by a Conway algebra is the Jones–Conway polynomial, [a2], [a5], [a4].
A nice example of a fourelement Conway algebra, which leads to the link invariant distinguishing the lefthanded and righthanded trefoil knots (cf. also Torus knot) is described below:
\begin{equation*} a _ { 1 } = 1 , a _ { 2 } = 2, \end{equation*}
\begin{equation*} a_3 = 4 ,\; a _ { i + 3} = a _ { i }. \end{equation*}
The operations $$ and $*$ are given by the following tables:


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, $w _ { L }$, yielded by (partial) Conway algebras have the properties that for the Conway skein triple $L _ { + }$, $L_{}$ and $L_0$:
\begin{equation*} w _ { L _ { + } } = w _ { L  }  w _ { L _ { 0 } }, \end{equation*}
\begin{equation*} w _ { L _ {  } } = w _ { L _ { + } } * w _ { L _ { 0 } } \end{equation*}
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=50906