The Kauffman Polynomial: Difference between revisions

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(here <math>s</math> is a strand and <math>s_\pm</math> is the same strand with a <math>\pm</math> kink added) and
(here <math>s</math> is a strand and <math>s_\pm</math> is the same strand with a <math>\pm</math> kink added) and


<center><math>L(T_1)+L(T_2) = z\left(L(T_3)+L(T_4)\right)</math></center>
<center><math>L(\backoverslash)+L(\slashoverback) = z\left(L(\smoothing)+L(\hsmoothing)\right)</math></center>


(here <math>T_1</math>, <math>T_2</math>, <math>T_3</math> and <math>T_4</math> are [[Image:backoverslash symbol.gif|20px]], [[Image:slashoverback symbol.gif|20px]], [[Image:vsmoothing symbol.gif|20px]] and [[Image:hsmoothing symbol.gif|20px]], respectively), and by the initial condition <math>L(U)=1</math> where <math>U</math> is the unknot [[Image:BigCirc symbol.gif|20px]].
and by the initial condition <math>L(U)=1</math> where <math>U</math> is the unknot [[Image:BigCirc symbol.gif|20px]].


<code>KnotTheory`</code> knows about the Kauffman polynomial:
<code>KnotTheory`</code> knows about the Kauffman polynomial:
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<!--$$?Kauffman$$-->
<!--$$?Kauffman$$-->
<!--Robot Land, no human edits to "END"-->
<!--Robot Land, no human edits to "END"-->
{{HelpAndAbout|
{{HelpAndAbout1|n=1|s=Kauffman}}
n = 2 |
⚫
Kauffman[K][a, z] computes the Kauffman polynomial of a knot or link K, in the variables a and z.
n1 = 3 |
{{HelpAndAbout2|n=2|s=Kauffman}}
in = <nowiki>Kauffman</nowiki> |
⚫
The Kauffman program was written by Scott Morrison.
⚫
out= <nowiki>Kauffman[K][a, z] computes the Kauffman polynomial of a knot or link K, in the variables a and z.</nowiki> |
{{HelpAndAbout3}}
⚫
about= <nowiki>The Kauffman polynomial program was written by Scott Morrison.</nowiki>}}
<!--END-->
<!--END-->


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<!--$$Kauffman[Knot[5, 2]][a, z]$$-->
<!--$$Kauffman[Knot[5, 2]][a, z]$$-->
<!--Robot Land, no human edits to "END"-->
<!--Robot Land, no human edits to "END"-->
{{InOut1|n=3}}
{{InOut|
n = 4 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Knot[5, 2]][a, z]</nowiki></pre>
in = <nowiki>Kauffman[Knot[5, 2]][a, z]</nowiki> |
{{InOut2|n=3}}<pre style="border: 0px; padding: 0em"><nowiki> 2 4 6 5 7 2 2 4 2 6 2 3 3
out= <nowiki> 2 4 6 5 7 2 2 4 2 6 2 3 3
-a + a + a - 2 a z - 2 a z + a z - a z - 2 a z + a z +
-a + a + a - 2 a z - 2 a z + a z - a z - 2 a z + a z +
5 3 7 3 4 4 6 4
5 3 7 3 4 4 6 4
2 a z + a z + a z + a z</nowiki></pre>
2 a z + a z + a z + a z</nowiki>}}
{{InOut3}}
<!--END-->
<!--END-->

{{Knot Image Pair|5_2|gif|T(8,3)|jpg}}


It is well known that the Jones polynomial is related to the Kauffman polynomial via
It is well known that the Jones polynomial is related to the Kauffman polynomial via


<center><math>J(L)(q) = (-1)^cL(K)(-q^{-3/4},\,q^{1/4}+q^{-1/4})</math>,</center>
<center><math>J(L)(q) = (-1)^{c+1}L(K)(-q^{-3/4},\,q^{1/4}+q^{-1/4})</math>,</center>


where <math>K</math> is some knot or link and where <math>c</math> is the number of components of <math>K</math>. Let us verify this fact for the torus knot [[T(8,3)]]:
where <math>K</math> is some knot or link and where <math>c</math> is the number of components of <math>K</math>. Let us verify this fact for the torus knot [[T(8,3)]]:
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<!--$$K = TorusKnot[8, 3];$$-->
<!--$$K = TorusKnot[8, 3];$$-->
<!--Robot Land, no human edits to "END"-->
<!--Robot Land, no human edits to "END"-->
{{In1|n=4}}
{{In|
n = 5 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>K = TorusKnot[8, 3];</nowiki></pre>
in = <nowiki>K = TorusKnot[8, 3];</nowiki>}}
{{In2}}
<!--END-->
<!--END-->


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}]$$-->
}]$$-->
<!--Robot Land, no human edits to "END"-->
<!--Robot Land, no human edits to "END"-->
{{InOut1|n=5}}
{{InOut|
n = 6 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>Simplify[{
in = <nowiki>Simplify[{
(-1)^(Length[Skeleton[K]]-1)Kauffman[K][-q^(-3/4), q^(1/4)+q^(-1/4)],
(-1)^(Length[Skeleton[K]]-1)Kauffman[K][-q^(-3/4), q^(1/4)+q^(-1/4)],
Jones[K][q]
Jones[K][q]
}]</nowiki></pre>
}]</nowiki> |
{{InOut2|n=5}}<pre style="border: 0px; padding: 0em"><nowiki> 7 9 16 7 9 16
out= <nowiki> 7 9 16 7 9 16
{q + q - q , q + q - q }</nowiki></pre>
{q + q - q , q + q - q }</nowiki>}}
{{InOut3}}
<!--END-->
<!--END-->


Latest revision as of 18:23, 21 February 2013


KnotTheory`/Navigation 

The Mathematica Package KnotTheory`

The Kauffman polynomial (see [Kauffman]) of a knot or link is where is the writhe of (see How is the Jones Polynomial Computed?) and where is the regular isotopy invariant defined by the skein relations

(here is a strand and is the same strand with a kink added) and

Failed to parse (unknown function "\backoverslash"): {\displaystyle L(\backoverslash)+L(\slashoverback) = z\left(L(\smoothing)+L(\hsmoothing)\right)}

and by the initial condition where is the unknot BigCirc symbol.gif.

KnotTheory` knows about the Kauffman polynomial:

(For In[1] see Setup)

In[2]:= ?Kauffman
Kauffman[K][a, z] computes the Kauffman polynomial of a knot or link K, in the variables a and z.
In[3]:= Kauffman::about
The Kauffman polynomial program was written by Scott Morrison.

Thus, for example, here's the Kauffman polynomial of the knot 5_2:

In[4]:= Kauffman[Knot[5, 2]][a, z]
Out[4]= 2 4 6 5 7 2 2 4 2 6 2 3 3 -a + a + a - 2 a z - 2 a z + a z - a z - 2 a z + a z + 5 3 7 3 4 4 6 4 2 a z + a z + a z + a z
5 2.gif
5_2 		T(8,3).jpg
T(8,3)

It is well known that the Jones polynomial is related to the Kauffman polynomial via

,

where is some knot or link and where is the number of components of . Let us verify this fact for the torus knot T(8,3):

In[5]:= K = TorusKnot[8, 3];
In[6]:= Simplify[{ (-1)^(Length[Skeleton[K]]-1)Kauffman[K][-q^(-3/4), q^(1/4)+q^(-1/4)], Jones[K][q] }]
Out[6]= 7 9 16 7 9 16 {q + q - q , q + q - q }

[Kauffman] ^  L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 312 (1990) 417-471.

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