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 quasi-group (cf. also Quasi-group). 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 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:

\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:

$|$ 1 2 3 4 1 2 1 4 3 2 3 4 1 2 3 1 2 3 4 4 4 3 2 1

 

$*$ 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, $w _ { L }$, 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