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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]], [[#References|[a2]]], [[#References|[a5]]], [[#References|[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:

<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellpadding="4" cellspacing="1" style="background-color:black;">  <tr> <td colname="1" colspan="1" style="background-color:white;">$|$</td> <td colname="2" colspan="1" style="background-color:white;">1</td> <td colname="3" colspan="1" style="background-color:white;">2</td> <td colname="4" colspan="1" style="background-color:white;">3</td> <td colname="5" colspan="1" style="background-color:white;">4</td> </tr> <tr> <td colname="1" colspan="1" style="background-color:white;"></td> <td colname="5" colspan="1" style="background-color:white;"></td> <td colname="4" colspan="1" style="background-color:white;"></td> <td colname="3" colspan="1" style="background-color:white;"></td> <td colname="2" colspan="1" style="background-color:white;"></td> </tr> <tr> <td colname="1" colspan="1" style="background-color:white;">1</td> <td colname="2" colspan="1" style="background-color:white;">2</td> <td colname="3" colspan="1" style="background-color:white;">1</td> <td colname="4" colspan="1" style="background-color:white;">4</td> <td colname="5" colspan="1" style="background-color:white;">3</td> </tr> <tr> <td colname="1" colspan="1" style="background-color:white;">2</td> <td colname="2" colspan="1" style="background-color:white;">3</td> <td colname="3" colspan="1" style="background-color:white;">4</td> <td colname="4" colspan="1" style="background-color:white;">1</td> <td colname="5" colspan="1" style="background-color:white;">2</td> </tr> <tr> <td colname="1" colspan="1" style="background-color:white;">3</td> <td colname="2" colspan="1" style="background-color:white;">1</td> <td colname="3" colspan="1" style="background-color:white;">2</td> <td colname="4" colspan="1" style="background-color:white;">3</td> <td colname="5" colspan="1" style="background-color:white;">4</td> </tr> <tr> <td colname="1" colspan="1" style="background-color:white;">4</td> <td colname="2" colspan="1" style="background-color:white;">4</td> <td colname="3" colspan="1" style="background-color:white;">3</td> <td colname="4" colspan="1" style="background-color:white;">2</td> <td colname="5" colspan="1" style="background-color:white;">1</td> </tr>  </table></td></tr> </table>

&nbsp;

<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellpadding="4" cellspacing="1" style="background-color:black;">  <tr> <td colname="1" colspan="1" style="background-color:white;">$*$</td> <td colname="2" colspan="1" style="background-color:white;">1</td> <td colname="3" colspan="1" style="background-color:white;">2</td> <td colname="4" colspan="1" style="background-color:white;">3</td> <td colname="5" colspan="1" style="background-color:white;">4</td> </tr> <tr> <td colname="1" colspan="1" style="background-color:white;"></td> <td colname="5" colspan="1" style="background-color:white;"></td> <td colname="4" colspan="1" style="background-color:white;"></td> <td colname="3" colspan="1" style="background-color:white;"></td> <td colname="2" colspan="1" style="background-color:white;"></td> </tr> <tr> <td colname="1" colspan="1" style="background-color:white;">1</td> <td colname="2" colspan="1" style="background-color:white;">3</td> <td colname="3" colspan="1" style="background-color:white;">1</td> <td colname="4" colspan="1" style="background-color:white;">2</td> <td colname="5" colspan="1" style="background-color:white;">4</td> </tr> <tr> <td colname="1" colspan="1" style="background-color:white;">2</td> <td colname="2" colspan="1" style="background-color:white;">1</td> <td colname="3" colspan="1" style="background-color:white;">3</td> <td colname="4" colspan="1" style="background-color:white;">4</td> <td colname="5" colspan="1" style="background-color:white;">2</td> </tr> <tr> <td colname="1" colspan="1" style="background-color:white;">3</td> <td colname="2" colspan="1" style="background-color:white;">2</td> <td colname="3" colspan="1" style="background-color:white;">4</td> <td colname="4" colspan="1" style="background-color:white;">3</td> <td colname="5" colspan="1" style="background-color:white;">1</td> </tr> <tr> <td colname="1" colspan="1" style="background-color:white;">4</td> <td colname="2" colspan="1" style="background-color:white;">4</td> <td colname="3" colspan="1" style="background-color:white;">2</td> <td colname="4" colspan="1" style="background-color:white;">1</td> <td colname="5" colspan="1" style="background-color:white;">3</td> </tr>  </table></td></tr> </table>

If one allows partial Conway algebras, one also gets the Murasugi signature and Tristram–Levine signature of links [[#References|[a3]]]. The skein calculus (cf. also [[Skein module|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|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====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  J.H. Conway, "An enumeration of knots and links"  J. Leech (ed.) , ''Computational Problems in Abstract Algebra'' , Pergamon  (1969)  pp. 329–358</td></tr>
<tr><td valign="top">[a2]</td> <td valign="top">  J.H. Przytycki, P. Traczyk,   "Invariants of links of Conway type"  ''Kobe J. Math.'' , '''4'''  (1987)  pp. 115–139</td></tr>
<tr><td valign="top">[a3]</td> <td valign="top">  J.H. Przytycki, P. Traczyk,   "Conway algebras and skein equivalence of links"  ''Proc. Amer. Math. Soc.'' , '''100''' :  4  (1987)  pp. 744–748</td></tr>
<tr><td valign="top">[a4]</td> <td valign="top">  A.S. Sikora, "On Conway algebras and the Homflypt polynomial"  ''J. Knot Th. Ramifications'' , '''6''' :  6  (1997)  pp. 879–893</td></tr>
<tr><td valign="top">[a5]</td> <td valign="top">  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</td></tr>
</table>