Namespaces
Variants
Actions

Conway algebra

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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
How to Cite This Entry:
Conway algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conway_algebra&oldid=55668
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article