Namespaces
Variants
Actions

Difference between revisions of "Jaeger composition product"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (AUTOMATIC EDIT (latexlist): Replaced 45 formulas out of 48 by TEX code with an average confidence of 2.0 and a minimal confidence of 2.0.)
m (more tex)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
 +
 
<!--This article has been texified automatically. Since there was no Nroff source code for this article,  
 
<!--This article has been texified automatically. Since there was no Nroff source code for this article,  
 
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 
was used.
 
was used.
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
+
If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
  
 
Out of 48 formulas, 45 were replaced by TEX code.-->
 
Out of 48 formulas, 45 were replaced by TEX code.-->
  
{{TEX|semi-auto}}{{TEX|partial}}
+
{{TEX|semi-auto}}{{TEX|part}}
 
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]]].
  
Line 15: Line 16:
 
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
 
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*}
+
\begin{equation*}
 +
Q _ { D _ { + } } - Q _ { D _ { - } } =  
 +
\left\{  
 +
\begin{array} { l }  
 +
{ Q _ { D _ { 0 } } } \text{ for a self 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 ([[#References|[a1]]]): Let $D$ be a diagram of an oriented link in $S ^ { 3 }$, 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
  
\begin{equation*} Q _ { D } ( v _ { 1 } v _ { 2 } , z ) = \sum _ { f \in \text{lbl} ( D ) } \end{equation*}
+
\begin{equation*}  
 
+
Q _ { D } ( v _ { 1 } v _ { 2 } , z ) = \sum _ { f \in \text{lbl} ( D ) }  
<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>
+
\langle D | f \rangle
 +
v_2^{\operatorname{rot}(D_{f,1})}  
 +
Q_{D_{f,1}}(v_1,z)
 +
v_1^{-\operatorname{rot}(D_{f,2})}
 +
Q_{D_{f,2}} (v_2,z).
 +
\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:
 
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:
Line 33: Line 46:
 
====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>
 +
 +
{{OldImage}}

Latest revision as of 05:30, 15 February 2024

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 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 ) } \langle D | f \rangle v_2^{\operatorname{rot}(D_{f,1})} Q_{D_{f,1}}(v_1,z) v_1^{-\operatorname{rot}(D_{f,2})} Q_{D_{f,2}} (v_2,z). \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


🛠️ This page contains images that should be replaced by better images in the SVG file format. 🛠️
How to Cite This Entry:
Jaeger composition product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jaeger_composition_product&oldid=50387
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article