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An abstract algebra which yields an invariant of links in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c1302101.png" /> (cf. also [[Link|Link]]).
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The concept is related to the entropic right quasi-group (cf. also [[Quasi-group|Quasi-group]]). A Conway algebra consists of a sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c1302102.png" />-argument operations (constants) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c1302103.png" /> and two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c1302104.png" />-argument operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c1302105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c1302106.png" />, which satisfy the following conditions:
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An abstract algebra which yields an invariant of links in $\mathbf{R}^{3}$ (cf. also [[Link]]).
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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:
 
Initial conditions:
  
C1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c1302107.png" />;
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C1) $a _ { n } | a _ {n  + 1} = a _ { n }$;
  
C2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c1302108.png" />.
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C2) $a _ { n } * a _ { n  + 1} = a _ { n }$.
  
 
Transposition properties:
 
Transposition properties:
  
C3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c1302109.png" />;
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C3) $( a | b ) | ( c | d ) = ( a | c ) | ( b | d )$;
  
C4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c13021010.png" />;
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C4) $( a | b ) * ( c | d ) = ( a  *  c ) | ( b  *  d )$;
  
C5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c13021011.png" />.
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C5) $( a * b ) * ( c * d ) = ( a * c ) * ( b * d )$.
  
 
Inverse operation properties:
 
Inverse operation properties:
  
C6) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c13021012.png" />;
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C6) $( a | b )  *  b  = a$;
  
C7) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c13021013.png" />. The main link invariant yielded by a Conway algebra is the [[Jones–Conway polynomial|Jones–Conway polynomial]], [[#References|[a2]]], [[#References|[a5]]], [[#References|[a4]]].
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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|Torus knot]]) is described below:
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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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c13021014.png" /></td> </tr></table>
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\begin{equation*} a_{1} = 1 , a_{2} = 2, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c13021015.png" /></td> </tr></table>
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\begin{equation*} a_3 = 4 ,\; a _ { i  + 3} = a _ { i }. \end{equation*}
  
The operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c13021016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c13021017.png" /> are given by the following tables:''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c13021018.png" /></td> <td colname="2" style="background-color:white;" colspan="1">1</td> <td colname="3" style="background-color:white;" colspan="1">2</td> <td colname="4" style="background-color:white;" colspan="1">3</td> <td colname="5" style="background-color:white;" colspan="1">4</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="5" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">1</td> <td colname="2" style="background-color:white;" colspan="1">2</td> <td colname="3" style="background-color:white;" colspan="1">1</td> <td colname="4" style="background-color:white;" colspan="1">4</td> <td colname="5" style="background-color:white;" colspan="1">3</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">2</td> <td colname="2" style="background-color:white;" colspan="1">3</td> <td colname="3" style="background-color:white;" colspan="1">4</td> <td colname="4" style="background-color:white;" colspan="1">1</td> <td colname="5" style="background-color:white;" colspan="1">2</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">3</td> <td colname="2" style="background-color:white;" colspan="1">1</td> <td colname="3" style="background-color:white;" colspan="1">2</td> <td colname="4" style="background-color:white;" colspan="1">3</td> <td colname="5" style="background-color:white;" colspan="1">4</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">4</td> <td colname="2" style="background-color:white;" colspan="1">4</td> <td colname="3" style="background-color:white;" colspan="1">3</td> <td colname="4" style="background-color:white;" colspan="1">2</td> <td colname="5" style="background-color:white;" colspan="1">1</td> </tr> </tbody> </table>
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The operations $|$ and $*$ are given by the following tables:
  
</td></tr> </table>
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<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>
  
''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c13021019.png" /></td> <td colname="2" style="background-color:white;" colspan="1">1</td> <td colname="3" style="background-color:white;" colspan="1">2</td> <td colname="4" style="background-color:white;" colspan="1">3</td> <td colname="5" style="background-color:white;" colspan="1">4</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="5" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">1</td> <td colname="2" style="background-color:white;" colspan="1">3</td> <td colname="3" style="background-color:white;" colspan="1">1</td> <td colname="4" style="background-color:white;" colspan="1">2</td> <td colname="5" style="background-color:white;" colspan="1">4</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">2</td> <td colname="2" style="background-color:white;" colspan="1">1</td> <td colname="3" style="background-color:white;" colspan="1">3</td> <td colname="4" style="background-color:white;" colspan="1">4</td> <td colname="5" style="background-color:white;" colspan="1">2</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">3</td> <td colname="2" style="background-color:white;" colspan="1">2</td> <td colname="3" style="background-color:white;" colspan="1">4</td> <td colname="4" style="background-color:white;" colspan="1">3</td> <td colname="5" style="background-color:white;" colspan="1">1</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">4</td> <td colname="2" style="background-color:white;" colspan="1">4</td> <td colname="3" style="background-color:white;" colspan="1">2</td> <td colname="4" style="background-color:white;" colspan="1">1</td> <td colname="5" style="background-color:white;" colspan="1">3</td> </tr> </tbody> </table>
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&nbsp;
  
</td></tr> </table>
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<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.
 
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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c13021020.png" />, yielded by (partial) Conway algebras have the properties that for the [[Conway skein triple|Conway skein triple]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c13021021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c13021022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c13021023.png" />:
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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 $L_0$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c13021024.png" /></td> </tr></table>
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\begin{equation*} w _ { L _ { + } } = w _ { L - } | w _ { L _ { 0 } }, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130210/c13021025.png" /></td> </tr></table>
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\begin{equation*} w _ { L _ { - } } = w _ { L _ { + } } * w _ { L _ { 0 } } \end{equation*}
  
 
====References====
 
====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>
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<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>

Latest revision as of 07:05, 24 March 2024

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=18395
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article