The Kauffman Polynomial

2008-12-16T16:30:35Z

EltleTorel: cboeltda

ounobovarro
{{Manual TOC Sidebar}}

The ''Kauffman polynomial'' <math>F(K)(a,z)</math> (see {{ref|Kauffman}}) of a knot or link <math>K</math> is <math>a^{-w(K)}L(K)</math> where <math>w(L)</math> is the writhe of <math>K</math> (see [[The_Jones_Polynomial#How_is_the_Jones_polynomial_computed.3F|How is the Jones Polynomial Computed?]]) and where <math>L(K)</math> is the regular isotopy invariant defined by the skein relations

<center><math>L(s_+)=aL(s), \qquad L(s_-)=a^{-1}L(s)</math></center>

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

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:

{{Startup Note}}



{{HelpAndAbout|
n = 2 |
n1 = 3 |
in = <nowiki>Kauffman</nowiki> |
out= <nowiki>Kauffman[K][a, z] computes the Kauffman polynomial of a knot or link K, in the variables a and z.</nowiki> |
about= <nowiki>The Kauffman program was written by Scott Morrison.</nowiki>}}


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


{{InOut|
n = 4 |
in = <nowiki>Kauffman[Knot[5, 2]][a, z]</nowiki> |
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 +

5 3 7 3 4 4 6 4
2 a z + a z + a z + a z</nowiki>}}


{{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

<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)]]:



{{In|
n = 5 |
in = <nowiki>K = TorusKnot[8, 3];</nowiki>}}




{{InOut|
n = 6 |
in = <nowiki>Simplify[{
(-1)^(Length[Skeleton[K]]-1)Kauffman[K][-q^(-3/4), q^(1/4)+q^(-1/4)],
Jones[K][q]
}]</nowiki> |
out= <nowiki> 7 9 16 7 9 16
{q + q - q , q + q - q }</nowiki>}}


{{note|Kauffman}} L. H. Kauffman, ''An invariant of regular isotopy'', Trans. Amer. Math. Soc. '''312''' (1990) 417-471.

Knot Atlas - User contributions [en]

The Kauffman Polynomial