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	Snake Relation
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TwistedSnakeRelation

Edit detail for TwistedSnakeRelation revision 4 of 4

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Editor: Bill Page Time: 2011/05/09 08:31:55 GMT-7
Note: better graphic

added:
or equivalently

$$
\scalebox{1} % Change this value to rescale the drawing.
{
\begin{pspicture}(0,-0.92)(5.82,0.92)
\psbezier[linewidth=0.04,linestyle=dashed,dash=0.16cm 0.16cm](5.8,-0.1)(5.8,-0.9)(5.0,-0.9)(5.0,-0.1)
\psbezier[linewidth=0.04](5.8,-0.1)(5.8,0.7)(5.0,0.7)(5.0,-0.1)
\psbezier[linewidth=0.04,linestyle=dashed,dash=0.16cm 0.16cm](4.2,0.7)(4.2,0.1)(5.0,0.5)(5.0,-0.1)
\psbezier[linewidth=0.04](5.0,0.1)(5.0,-0.5)(4.2,-0.1)(4.2,-0.7)
\psbezier[linewidth=0.04,linestyle=dashed,dash=0.16cm 0.16cm](0.0,-0.1)(0.0,-0.9)(0.8,-0.9)(0.8,-0.1)
\psbezier[linewidth=0.04](0.0,-0.1)(0.0,0.7)(0.8,0.7)(0.8,-0.1)
\psbezier[linewidth=0.04,linestyle=dashed,dash=0.16cm 0.16cm](1.6,0.7)(1.6,0.1)(0.8,0.5)(0.8,-0.1)
\psbezier[linewidth=0.04](0.8,0.1)(0.8,-0.5)(1.6,-0.1)(1.6,-0.7)
\psline[linewidth=0.04cm](2.8,0.3)(2.8,-0.9)
\psline[linewidth=0.04cm](2.8,0.9)(2.8,0.3)
\psline[linewidth=0.04cm](4.2,0.9)(4.2,0.7)
\psline[linewidth=0.04cm](4.2,-0.9)(4.2,-0.7)
\psline[linewidth=0.04cm](1.6,0.9)(1.6,0.7)
\psline[linewidth=0.04cm](1.6,-0.7)(1.6,-0.9)
\usefont{T1}{ptm}{m}{n}
\rput(2.0948439,0.005){=}
\usefont{T1}{ptm}{m}{n}
\rput(3.4948437,0.005){=}
\end{pspicture} 
}
$$

Non-degeneracy of the pairing (snake relation)

$\scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.22)(5.82,1.22) \psbezier[linewidth=0.04](0.0,-0.4)(0.0,-1.2)(0.8,-1.2)(0.8,-0.4) \psbezier[linewidth=0.04](0.8,0.4)(0.8,1.2)(1.6,1.2)(1.6,0.4) \psline[linewidth=0.04cm](0.0,-0.4)(0.0,0.2) \psline[linewidth=0.04cm](1.6,-0.4)(1.6,-1.0) \psline[linewidth=0.04cm](2.8,0.2)(2.8,-1.0) \psbezier[linewidth=0.04](5.8,-0.4)(5.8,-1.2)(5.0,-1.2)(5.0,-0.4) \psbezier[linewidth=0.04](5.0,0.4)(5.0,1.2)(4.2,1.2)(4.2,0.4) \psline[linewidth=0.04cm](5.8,-0.4)(5.8,0.2) \psline[linewidth=0.04cm](4.2,-0.4)(4.2,-1.0) \usefont{T1}{ptm}{m}{n} \rput(2.0948439,-0.095){=} \usefont{T1}{ptm}{m}{n} \rput(3.4948437,-0.095){=} \psbezier[linewidth=0.04,linestyle=dashed,dash=0.16cm 0.16cm](4.2,0.4)(4.2,-0.2)(5.0,0.2)(5.0,-0.4) \psbezier[linewidth=0.04](5.0,0.4)(5.0,-0.2)(4.2,0.2)(4.2,-0.4) \psbezier[linewidth=0.04,linestyle=dashed,dash=0.16cm 0.16cm](0.8,0.4)(0.8,-0.2)(1.6,0.2)(1.6,-0.4) \psbezier[linewidth=0.04](1.6,0.4)(1.6,-0.2)(0.8,0.2)(0.8,-0.4) \psline[linewidth=0.04cm](2.8,1.0)(2.8,0.2) \psline[linewidth=0.04cm](5.8,1.0)(5.8,0.2) \psline[linewidth=0.04cm](0.0,1.0)(0.0,0.2) \end{pspicture} }$

or equivalently

$\scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-0.92)(5.82,0.92) \psbezier[linewidth=0.04,linestyle=dashed,dash=0.16cm 0.16cm](5.8,-0.1)(5.8,-0.9)(5.0,-0.9)(5.0,-0.1) \psbezier[linewidth=0.04](5.8,-0.1)(5.8,0.7)(5.0,0.7)(5.0,-0.1) \psbezier[linewidth=0.04,linestyle=dashed,dash=0.16cm 0.16cm](4.2,0.7)(4.2,0.1)(5.0,0.5)(5.0,-0.1) \psbezier[linewidth=0.04](5.0,0.1)(5.0,-0.5)(4.2,-0.1)(4.2,-0.7) \psbezier[linewidth=0.04,linestyle=dashed,dash=0.16cm 0.16cm](0.0,-0.1)(0.0,-0.9)(0.8,-0.9)(0.8,-0.1) \psbezier[linewidth=0.04](0.0,-0.1)(0.0,0.7)(0.8,0.7)(0.8,-0.1) \psbezier[linewidth=0.04,linestyle=dashed,dash=0.16cm 0.16cm](1.6,0.7)(1.6,0.1)(0.8,0.5)(0.8,-0.1) \psbezier[linewidth=0.04](0.8,0.1)(0.8,-0.5)(1.6,-0.1)(1.6,-0.7) \psline[linewidth=0.04cm](2.8,0.3)(2.8,-0.9) \psline[linewidth=0.04cm](2.8,0.9)(2.8,0.3) \psline[linewidth=0.04cm](4.2,0.9)(4.2,0.7) \psline[linewidth=0.04cm](4.2,-0.9)(4.2,-0.7) \psline[linewidth=0.04cm](1.6,0.9)(1.6,0.7) \psline[linewidth=0.04cm](1.6,-0.7)(1.6,-0.9) \usefont{T1}{ptm}{m}{n} \rput(2.0948439,0.005){=} \usefont{T1}{ptm}{m}{n} \rput(3.4948437,0.005){=} \end{pspicture} }$

Ref:

http://mat.uab.es/~kock/TQFT.html
Frobenius algebras and 2D topological quantum field theories

Section 2.3.11, page 112.
Also related notes: http://mat.uab.es/~kock/TQFT/FS.pdf
Section 2.2.9, page 23.

Joachim Kock
http://arxiv.org/abs/math/0512103
Categorical Aspects of Topological Quantum Field Theories

Section 2.3.3, page 27.

Bruce H. Bartlett

We use the Axiom LinearOperator library

fricas

(1) -> )library CARTEN MONAL PROP LOP

   >> System error:
   The value
  15684
is not of type
  LIST

and some convenient notation

fricas

macro Σ(x,i,n)==reduce(+,[x for i in n])

Type: Void

fricas

macro Ξ(f,i,n)==[f for i in n]

Type: Void

fricas

macro sb == subscript

Type: Void

fricas

macro sp == superscript

Type: Void

Let 𝐋 be the domain of 2-dimensional linear operators

fricas

dim:=2

$\label{eq1}2$

(1)

Type: PositiveInteger?

fricas

macro ℒ == List

Type: Void

fricas

macro ℚ == Expression Integer

Type: Void

fricas

𝐋 := LinearOperator(OVAR ['1,'2], ℚ)

   There are no library operations named LinearOperator 
      Use HyperDoc Browse or issue
                           )what op LinearOperator
      to learn if there is any operation containing " LinearOperator " 
      in its name.

   Cannot find a definition or applicable library operation named 
      LinearOperator with argument type(s) 
                                    Type
                                    Type

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Pairing

A scalar product (pairing) is represented by

fricas

U:=Σ(Σ(sp('u,[i,j])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)

   There are no library operations named 𝐝 
      Use HyperDoc Browse or issue
                                 )what op 𝐝
      to learn if there is any operation containing " 𝐝 " in its name.
   Cannot find a definition or applicable library operation named 𝐝 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are no library operations named 𝐝 
      Use HyperDoc Browse or issue
                                 )what op 𝐝
      to learn if there is any operation containing " 𝐝 " in its name.

   Cannot find a definition or applicable library operation named 𝐝 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

In general we do not require that it be symmetric.

Co-pairing

Solve the "twisted snake relation" as a system of linear equations.

fricas

Ω:𝐋:=Σ(Σ(sb('u,[i,j])*𝐞.i*𝐞.j, i,1..dim), j,1..dim)

   𝐋 is not a valid type.

This is equivalent to a matrix inverse (transposed!)

fricas

Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/U, i,1..dim), j,1..dim)

   There are no library operations named 𝐞 
      Use HyperDoc Browse or issue
                                 )what op 𝐞
      to learn if there is any operation containing " 𝐞 " in its name.
   Cannot find a definition or applicable library operation named 𝐞 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are no library operations named 𝐞 
      Use HyperDoc Browse or issue
                                 )what op 𝐞
      to learn if there is any operation containing " 𝐞 " in its name.

   Cannot find a definition or applicable library operation named 𝐞 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Check that the twisted snake relation holds

fricas

test
  (  I Ω   )  /
  (  I X   )  /
  (   U I  )  =  I

   There are no exposed library operations named I but there is one 
      unexposed operation with that name. Use HyperDoc Browse or issue
                                )display op I
      to learn more about the available operation.

   Cannot find a definition or applicable library operation named I 
      with argument type(s) 
                                 Variable(Ω)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Dimension

Since the "snake" is twisted, dimension is as expected.

$\scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-0.92)(0.82,0.92) \psbezier[linewidth=0.04](0.0,-0.1)(0.0,-0.9)(0.8,-0.9)(0.8,-0.1) \psbezier[linewidth=0.04](0.0,0.1)(0.0,0.9)(0.8,0.9)(0.8,0.1) \psline[linewidth=0.04cm](0.0,0.1)(0.0,-0.1) \psline[linewidth=0.04cm](0.8,0.1)(0.8,-0.1) \end{pspicture} }$

fricas

d:=
    Ω /
    U

$\label{eq2}\frac{��}{U}$

(2)

Type: Fraction(Polynomial(Integer))

This "twisted dimension " depends on !

$\scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.22)(0.82,1.22) \psbezier[linewidth=0.04](0.0,-0.4)(0.0,-1.2)(0.8,-1.2)(0.8,-0.4) \psbezier[linewidth=0.04](0.0,0.4)(0.0,1.2)(0.8,1.2)(0.8,0.4) \psbezier[linewidth=0.04,linestyle=dashed,dash=0.16cm 0.16cm](0.0,0.4)(0.0,-0.2)(0.8,0.2)(0.8,-0.4) \psbezier[linewidth=0.04](0.8,0.4)(0.8,-0.2)(0.0,0.2)(0.0,-0.4) \end{pspicture} }$

fricas

d':=
     Ω /
     X /
     U

$\label{eq3}\frac{��}{U \ X}$

(3)

Type: Fraction(Polynomial(Integer))