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A formula for the [[Jones–Conway polynomial|Jones–Conway polynomial]], describing it as a sum of products of the Jones–Conway polynomials of pieces of the diagram. It has its root in the statistical mechanics model of the Jones–Conway polynomial by V.F.R. Jones. It has been applied to periodic links and to the building of a [[Hopf algebra|Hopf algebra]] structure on the Jones–Conway [[Skein module|skein module]] of the product of a surface and an interval [[#References|[a3]]], [[#References|[a4]]], [[#References|[a2]]].
 
A formula for the [[Jones–Conway polynomial|Jones–Conway polynomial]], describing it as a sum of products of the Jones–Conway polynomials of pieces of the diagram. It has its root in the statistical mechanics model of the Jones–Conway polynomial by V.F.R. Jones. It has been applied to periodic links and to the building of a [[Hopf algebra|Hopf algebra]] structure on the Jones–Conway [[Skein module|skein module]] of the product of a surface and an interval [[#References|[a3]]], [[#References|[a4]]], [[#References|[a2]]].
  
 
To define the Jaeger composition product it is convenient to work with the following regular isotopy variant of the Jones–Conway polynomial:
 
To define the Jaeger composition product it is convenient to work with the following regular isotopy variant of the Jones–Conway polynomial:
  
<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/j/j130/j130010/j1300101.png" /></td> </tr></table>
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\begin{equation*} Q _ { D } ( v , z ) = z ^ { \operatorname { com } ( D ) - 1 } v ^ { - \operatorname { Tait } ( D ) } ( v ^ { - 1 } - v ) P _ { D } ( v , z ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j1300102.png" /> is the number of link components and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j1300103.png" /> is the algebraic sum of the signs of the crossings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j1300104.png" />. It is also convenient to add the empty link, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j1300105.png" />, to the set of links and put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j1300106.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j1300107.png" /> satisfies the skein relation
+
where $\operatorname { com }( D )$ is the number of link components and $\operatorname{Tait}( D )$ is the algebraic sum of the signs of the crossings of $D$. It is also convenient to add the empty link, $\emptyset$, to the set of links and put $Q _ { \emptyset } ( v , z ) = 1$. $Q _ { D } ( v , z )$ satisfies the skein relation
  
<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/j/j130/j130010/j1300108.png" /></td> </tr></table>
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\begin{equation*} Q _ { D _ { + } } - Q _ { D _ { - } } = \left\{ \begin{array} { l } { Q _ { D _ { 0 } } } \text{ for a self}\square\text{ crossing}, \\ { z ^ { 2 } Q _ { D _ { 0 } } }\text{ for a mixed crossing}, \end{array} \right. \end{equation*}
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j1300109.png" />. The advantage of working with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001010.png" /> is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001011.png" /> (no negative powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001012.png" />) and that the Jaeger composition product has a nice simple form. Indeed ([[#References|[a1]]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001013.png" /> be a diagram of an oriented link in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001014.png" />, then
+
and $Q _ { D \cup 0 } = ( v ^ { - 1 } - v ) Q _ { D }$. The advantage of working with $Q _ { D } ( v , z )$ is that $Q _ { D } ( v , z ) \in \mathbf{Z} [ v ^ { \pm 1 } , z ^ { 2 } ]$ (no negative powers of $z$) and that the Jaeger composition product has a nice simple form. Indeed ([[#References|[a1]]]): Let $D$ be a diagram of an oriented link in $S ^ { 3 }$, then
  
<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/j/j130/j130010/j13001015.png" /></td> </tr></table>
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\begin{equation*} Q _ { D } ( v _ { 1 } v _ { 2 } , z ) = \sum _ { f \in \text{lbl} ( D ) } \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/j/j130/j130010/j13001016.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001016.png"/></td> </tr></table>
  
The meaning of the used symbols is as follows. To define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001017.png" />, consider <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001018.png" /> as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001019.png" />-valent graph. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001020.png" /> denote the set of edges of the graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001021.png" />. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001023.png" />-labelling of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001024.png" /> is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001025.png" /> such that around a vertex the following labellings are allowed:
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The meaning of the used symbols is as follows. To define $\operatorname{lbl} ( D )$, consider $D$ as a $4$-valent graph. Let $\operatorname{Edge}( D )$ denote the set of edges of the graph $D$. A $2$-labelling of $D$ is a function $f : \text { Edge } ( D ) \rightarrow \{ 1,2 \}$ such that around a vertex the following labellings are allowed:
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/j130010a.gif" />
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<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/j130010a.gif" style="border:1px solid;"/>
  
 
Figure: j130010a
 
Figure: j130010a
  
The set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001026.png" />-labellings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001027.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001028.png" />. The edges of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001029.png" /> with label <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001030.png" /> form an oriented link diagram, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001031.png" />. The vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001032.png" /> which are neither in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001033.png" /> nor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001034.png" /> are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001036.png" />-smoothing vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001037.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001038.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001039.png" />) denote the number of negative (respectively, positive) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001040.png" />-smoothing vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001041.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001042.png" /> and define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001043.png" />. Finally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001044.png" /> denotes the rotational number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001045.png" />, i.e. the sum of the signs of the Seifert circles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001046.png" />, where the sign of such a circle is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001047.png" /> if it is oriented counterclockwise and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130010/j13001048.png" /> otherwise.
+
The set of $2$-labellings of $D$ is denoted by $\operatorname{lbl} ( D )$. The edges of $D$ with label $i$ form an oriented link diagram, denoted by $D _ { f , i }$. The vertices of $D$ which are neither in $D _{f , 1}$ nor $D_{f , 2}$ are called $f$-smoothing vertices of $D$. Let $| f | _ { - }$ (respectively, $| f |_{ +}$) denote the number of negative (respectively, positive) $f$-smoothing vertices of $D$. Let $| f | = | f |_{ -} + | f |_{+}$ and define $\langle D | f \rangle = ( - 1 ) ^ { | f | } -  z ^{ | f | - \operatorname { com } ( D _ { f , 1 } ) - \operatorname { com } ( D _ { f , 2 } ) + \operatorname { com } ( D )}$. Finally, $\operatorname{rot} ( D )$ denotes the rotational number of $D$, i.e. the sum of the signs of the Seifert circles of $D$, where the sign of such a circle is $1$ if it is oriented counterclockwise and $- 1$ otherwise.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Jaeger,  "Composition products and models for the Homfly polynomial"  ''L'Enseign. Math.'' , '''35'''  (1989)  pp. 323–361</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.H. Przytycki,  "Quantum group of links in a handlebody"  M. Gerstenhaber (ed.)  J.D. Stasheff (ed.) , ''Deformation Theory and Quantum Groups with Applications to Mathematical Physics'' , ''Contemp. Math.'' , '''134'''  (1992)  pp. 235–245</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.H. Przytycki,  "A simple proof of the Traczyk–Yokota criteria for periodic knots"  ''Proc. Amer. Math. Soc.'' , '''123'''  (1995)  pp. 1607–1611</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  V.G. Turaev,  "Skein quantization of Poisson algebras of loops on surfaces"  ''Ann. Sci. Ecole Norm. Sup.'' , '''4''' :  24  (1991)  pp. 635–704</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  F. Jaeger,  "Composition products and models for the Homfly polynomial"  ''L'Enseign. Math.'' , '''35'''  (1989)  pp. 323–361</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J.H. Przytycki,  "Quantum group of links in a handlebody"  M. Gerstenhaber (ed.)  J.D. Stasheff (ed.) , ''Deformation Theory and Quantum Groups with Applications to Mathematical Physics'' , ''Contemp. Math.'' , '''134'''  (1992)  pp. 235–245</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J.H. Przytycki,  "A simple proof of the Traczyk–Yokota criteria for periodic knots"  ''Proc. Amer. Math. Soc.'' , '''123'''  (1995)  pp. 1607–1611</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  V.G. Turaev,  "Skein quantization of Poisson algebras of loops on surfaces"  ''Ann. Sci. Ecole Norm. Sup.'' , '''4''' :  24  (1991)  pp. 635–704</td></tr></table>

Latest revision as of 17:46, 1 July 2020

A formula for the Jones–Conway polynomial, describing it as a sum of products of the Jones–Conway polynomials of pieces of the diagram. It has its root in the statistical mechanics model of the Jones–Conway polynomial by V.F.R. Jones. It has been applied to periodic links and to the building of a Hopf algebra structure on the Jones–Conway skein module of the product of a surface and an interval [a3], [a4], [a2].

To define the Jaeger composition product it is convenient to work with the following regular isotopy variant of the Jones–Conway polynomial:

\begin{equation*} Q _ { D } ( v , z ) = z ^ { \operatorname { com } ( D ) - 1 } v ^ { - \operatorname { Tait } ( D ) } ( v ^ { - 1 } - v ) P _ { D } ( v , z ), \end{equation*}

where $\operatorname { com }( D )$ is the number of link components and $\operatorname{Tait}( D )$ is the algebraic sum of the signs of the crossings of $D$. It is also convenient to add the empty link, $\emptyset$, to the set of links and put $Q _ { \emptyset } ( v , z ) = 1$. $Q _ { D } ( v , z )$ satisfies the skein relation

\begin{equation*} Q _ { D _ { + } } - Q _ { D _ { - } } = \left\{ \begin{array} { l } { Q _ { D _ { 0 } } } \text{ for a self}\square\text{ crossing}, \\ { z ^ { 2 } Q _ { D _ { 0 } } }\text{ for a mixed crossing}, \end{array} \right. \end{equation*}

and $Q _ { D \cup 0 } = ( v ^ { - 1 } - v ) Q _ { D }$. The advantage of working with $Q _ { D } ( v , z )$ is that $Q _ { D } ( v , z ) \in \mathbf{Z} [ v ^ { \pm 1 } , z ^ { 2 } ]$ (no negative powers of $z$) and that the Jaeger composition product has a nice simple form. Indeed ([a1]): Let $D$ be a diagram of an oriented link in $S ^ { 3 }$, then

\begin{equation*} Q _ { D } ( v _ { 1 } v _ { 2 } , z ) = \sum _ { f \in \text{lbl} ( D ) } \end{equation*}

The meaning of the used symbols is as follows. To define $\operatorname{lbl} ( D )$, consider $D$ as a $4$-valent graph. Let $\operatorname{Edge}( D )$ denote the set of edges of the graph $D$. A $2$-labelling of $D$ is a function $f : \text { Edge } ( D ) \rightarrow \{ 1,2 \}$ such that around a vertex the following labellings are allowed:

Figure: j130010a

The set of $2$-labellings of $D$ is denoted by $\operatorname{lbl} ( D )$. The edges of $D$ with label $i$ form an oriented link diagram, denoted by $D _ { f , i }$. The vertices of $D$ which are neither in $D _{f , 1}$ nor $D_{f , 2}$ are called $f$-smoothing vertices of $D$. Let $| f | _ { - }$ (respectively, $| f |_{ +}$) denote the number of negative (respectively, positive) $f$-smoothing vertices of $D$. Let $| f | = | f |_{ -} + | f |_{+}$ and define $\langle D | f \rangle = ( - 1 ) ^ { | f | } - z ^{ | f | - \operatorname { com } ( D _ { f , 1 } ) - \operatorname { com } ( D _ { f , 2 } ) + \operatorname { com } ( D )}$. Finally, $\operatorname{rot} ( D )$ denotes the rotational number of $D$, i.e. the sum of the signs of the Seifert circles of $D$, where the sign of such a circle is $1$ if it is oriented counterclockwise and $- 1$ otherwise.

References

[a1] F. Jaeger, "Composition products and models for the Homfly polynomial" L'Enseign. Math. , 35 (1989) pp. 323–361
[a2] J.H. Przytycki, "Quantum group of links in a handlebody" M. Gerstenhaber (ed.) J.D. Stasheff (ed.) , Deformation Theory and Quantum Groups with Applications to Mathematical Physics , Contemp. Math. , 134 (1992) pp. 235–245
[a3] J.H. Przytycki, "A simple proof of the Traczyk–Yokota criteria for periodic knots" Proc. Amer. Math. Soc. , 123 (1995) pp. 1607–1611
[a4] V.G. Turaev, "Skein quantization of Poisson algebras of loops on surfaces" Ann. Sci. Ecole Norm. Sup. , 4 : 24 (1991) pp. 635–704
How to Cite This Entry:
Jaeger composition product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jaeger_composition_product&oldid=18470
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article