The Kauffman Polynomial - Revision history

ScottDataRobot at 22:23, 21 February 2013

2013-02-21T22:23:03Z

← Older revision		Revision as of 18:23, 21 February 2013
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	in = <nowiki>Kauffman</nowiki> \|		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> \|		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>}}		about= <nowiki>The Kauffman polynomial program was written by Scott Morrison.</nowiki>}}
	<!--END-->		<!--END-->

Drorbn: Reverted edits by EltleTorel (Talk); changed back to last version by Scott

2008-12-17T11:44:02Z

Reverted edits by EltleTorel (Talk); changed back to last version by Scott

← Older revision		Revision as of 07:44, 17 December 2008
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	ounobovarro
	{{Manual TOC Sidebar}}		{{Manual TOC Sidebar}}

EltleTorel: cboeltda

2008-12-16T16:30:35Z

cboeltda

← Older revision		Revision as of 12:30, 16 December 2008
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			ounobovarro
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Scott: fixing mistake in sign exponent. cf. email of October 14.

2008-10-14T08:30:01Z

fixing mistake in sign exponent. cf. email of October 14.

← Older revision		Revision as of 04:30, 14 October 2008
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	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)]]:

Scott: Reverted edits by 121.67.235.133 (Talk); changed back to last version by Drorbn

2007-08-28T23:30:50Z

Reverted edits by 121.67.235.133 (Talk); changed back to last version by Drorbn

← Older revision		Revision as of 19:30, 28 August 2007
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	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		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>		<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		(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>		<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]].		and by the initial condition <math>L(U)=1</math> where <math>U</math> is the unknot [[Image:BigCirc symbol.gif\|20px]].
Line 32:		Line 32:
	in = <nowiki>Kauffman[Knot[5, 2]][a, z]</nowiki> \|		in = <nowiki>Kauffman[Knot[5, 2]][a, z]</nowiki> \|
	out= <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>}}		2 a z + a z + a z + a z</nowiki>}}
	<!--END-->		<!--END-->

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	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)^cL(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)]]:
Line 54:		Line 54:

	<!--$$Simplify[{		<!--$$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]
	}]$$-->		}]$$-->
Line 61:		Line 61:
	n = 6 \|		n = 6 \|
	in = <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> \|		}]</nowiki> \|
	out= <nowiki> 7 9 16 7 9 16		out= <nowiki> 7 9 16 7 9 16
	{q q - q , q q - q }</nowiki>}}		{q + q - q , q + q - q }</nowiki>}}
	<!--END-->		<!--END-->

121.67.235.133 at 15:49, 28 August 2007

2007-08-28T15:49:57Z

← Older revision		Revision as of 11:49, 28 August 2007
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	{{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		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>		<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		(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>		<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]].		and by the initial condition <math>L(U)=1</math> where <math>U</math> is the unknot [[Image:BigCirc symbol.gif\|20px]].
Line 32:		Line 32:
	in = <nowiki>Kauffman[Knot[5, 2]][a, z]</nowiki> \|		in = <nowiki>Kauffman[Knot[5, 2]][a, z]</nowiki> \|
	out= <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>}}		2 a z a z a z a z</nowiki>}}
	<!--END-->		<!--END-->

Line 42:		Line 42:
	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)^cL(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)]]:
Line 54:		Line 54:

	<!--$$Simplify[{		<!--$$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]
	}]$$-->		}]$$-->
Line 61:		Line 61:
	n = 6 \|		n = 6 \|
	in = <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> \|		}]</nowiki> \|
	out= <nowiki> 7 9 16 7 9 16		out= <nowiki> 7 9 16 7 9 16
	{q + q - q , q + q - q }</nowiki>}}		{q q - q , q q - q }</nowiki>}}
	<!--END-->		<!--END-->

Drorbn: Reverted edit of 218.220.46.56, changed back to last version by Drorbn

2007-06-15T06:10:22Z

Reverted edit of 218.220.46.56, changed back to last version by Drorbn

← Older revision		Revision as of 02:10, 15 June 2007
Line 3:		Line 3:
	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		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>		<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		(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>		<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]].		and by the initial condition <math>L(U)=1</math> where <math>U</math> is the unknot [[Image:BigCirc symbol.gif\|20px]].
Line 32:		Line 32:
	in = <nowiki>Kauffman[Knot[5, 2]][a, z]</nowiki> \|		in = <nowiki>Kauffman[Knot[5, 2]][a, z]</nowiki> \|
	out= <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>}}		2 a z + a z + a z + a z</nowiki>}}
	<!--END-->		<!--END-->

Line 42:		Line 42:
	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)^cL(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)]]:
Line 54:		Line 54:

	<!--$$Simplify[{		<!--$$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]
	}]$$-->		}]$$-->
Line 61:		Line 61:
	n = 6 \|		n = 6 \|
	in = <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> \|		}]</nowiki> \|
	out= <nowiki> 7 9 16 7 9 16		out= <nowiki> 7 9 16 7 9 16
	{q q - q , q q - q }</nowiki>}}		{q + q - q , q + q - q }</nowiki>}}
	<!--END-->		<!--END-->

218.220.46.56 at 04:06, 15 June 2007

2007-06-15T04:06:49Z

← Older revision		Revision as of 00:06, 15 June 2007
Line 3:		Line 3:
	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		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>		<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		(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>		<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]].		and by the initial condition <math>L(U)=1</math> where <math>U</math> is the unknot [[Image:BigCirc symbol.gif\|20px]].
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	in = <nowiki>Kauffman[Knot[5, 2]][a, z]</nowiki> \|		in = <nowiki>Kauffman[Knot[5, 2]][a, z]</nowiki> \|
	out= <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>}}		2 a z a z a z a z</nowiki>}}
	<!--END-->		<!--END-->

Line 42:		Line 42:
	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)^cL(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)]]:
Line 54:		Line 54:

	<!--$$Simplify[{		<!--$$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]
	}]$$-->		}]$$-->
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	n = 6 \|		n = 6 \|
	in = <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> \|		}]</nowiki> \|
	out= <nowiki> 7 9 16 7 9 16		out= <nowiki> 7 9 16 7 9 16
	{q + q - q , q + q - q }</nowiki>}}		{q q - q , q q - q }</nowiki>}}
	<!--END-->		<!--END-->

Drorbn: Reverted edit of 66.150.105.20, changed back to last version by DrorsRobot

2007-04-12T10:39:53Z

Reverted edit of 66.150.105.20, changed back to last version by DrorsRobot

← Older revision		Revision as of 06:39, 12 April 2007
Line 3:		Line 3:
	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		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>		<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		(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>		<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]].		and by the initial condition <math>L(U)=1</math> where <math>U</math> is the unknot [[Image:BigCirc symbol.gif\|20px]].
Line 32:		Line 32:
	in = <nowiki>Kauffman[Knot[5, 2]][a, z]</nowiki> \|		in = <nowiki>Kauffman[Knot[5, 2]][a, z]</nowiki> \|
	out= <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>}}		2 a z + a z + a z + a z</nowiki>}}
	<!--END-->		<!--END-->

Line 42:		Line 42:
	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)^cL(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)]]:
Line 54:		Line 54:

	<!--$$Simplify[{		<!--$$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]
	}]$$-->		}]$$-->
Line 61:		Line 61:
	n = 6 \|		n = 6 \|
	in = <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> \|		}]</nowiki> \|
	out= <nowiki> 7 9 16 7 9 16		out= <nowiki> 7 9 16 7 9 16
	{q q - q , q q - q }</nowiki>}}		{q + q - q , q + q - q }</nowiki>}}
	<!--END-->		<!--END-->

66.150.105.20 at 04:34, 12 April 2007

2007-04-12T04:34:50Z

← Older revision		Revision as of 00:34, 12 April 2007
Line 3:		Line 3:
	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		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>		<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		(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>		<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]].		and by the initial condition <math>L(U)=1</math> where <math>U</math> is the unknot [[Image:BigCirc symbol.gif\|20px]].
Line 32:		Line 32:
	in = <nowiki>Kauffman[Knot[5, 2]][a, z]</nowiki> \|		in = <nowiki>Kauffman[Knot[5, 2]][a, z]</nowiki> \|
	out= <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>}}		2 a z a z a z a z</nowiki>}}
	<!--END-->		<!--END-->

Line 42:		Line 42:
	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)^cL(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)]]:
Line 54:		Line 54:

	<!--$$Simplify[{		<!--$$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]
	}]$$-->		}]$$-->
Line 61:		Line 61:
	n = 6 \|		n = 6 \|
	in = <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> \|		}]</nowiki> \|
	out= <nowiki> 7 9 16 7 9 16		out= <nowiki> 7 9 16 7 9 16
	{q + q - q , q + q - q }</nowiki>}}		{q q - q , q q - q }</nowiki>}}
	<!--END-->		<!--END-->