VasCalc Documentation - User's Guide - Revision history

Sankaran at 00:53, 13 September 2006

2006-09-13T00:53:02Z

← Older revision		Revision as of 20:53, 12 September 2006
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	«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]»}]</nowiki>}}		«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]»}]</nowiki>}}

	The invariant of the unknot <math>Z(O)</math> is given by closing <math>Z_\infty</math> to a circle ~~(the first~~ ~~line~~ ~~below),~~ and ~~taking~~ the ~~inverse~~:		The invariant of the unknot <math>Z(O)</math> is given by closing <math>Z_\infty^{-1}</math> to a circle; we then reduce and display the result:
	{{InOut\|n=12\|in=<nowiki>~~ZinfInverseCircle = ASeries~~[0,1,~~CD[(~~Zinf)^(-1)] /. ~~Line~~ -> ~~Circle]~~</nowiki>\|		{{InOut\|n=12\|in=<nowiki>ConnectStrand[1, 1, Zinf^(-1)] // ReduceFI // CD</nowiki>
		⚫	\|out = <nowiki>1 - CD[Circle[1, 2, 1, 2]]/24 - CD[Circle[1, 2, 3, 1, 4, 2, 3, 4]]/360
	out = <nowiki>ASeries[0, 1, {«JavaObject[vectorSpace.Coefficient]»,
		⚫	+ CD[Circle[1, 2, 3, 1, 4, 3, 2, 4]]/720 +
	«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]»,
			(7*CD[Circle[1, 2, 3, 4, 1, 2, 3, 4]])/5760</nowiki>}}
	«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]»}]</nowiki>}}
	{{InOut\|n=13\|in=<nowiki>CD[ReduceFI[ZinfInverseCircle]]</nowiki>\|
⚫	out=<nowiki>1 - CD[Circle[1, 2, 1, 2]]/24 - CD[Circle[1, 2, 3, 1, 4, 2, 3, 4]]/360 +
⚫	CD[Circle[1, 2, 3, 1, 4, 3, 2, 4]]/720 + ~~(7*CD[Circle[1, 2, 3, 4, 1, 2, 3, 4]])/5760 </nowiki>}}~~

	====The Kontsevich integral for unframed knots====		====The Kontsevich integral for unframed knots====

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	Here is the Kontsevich integral, up to degree four, of the trefoil:		Here is the Kontsevich integral, up to degree four, of the trefoil:
	{{InOut\|n=14\|in=<nowiki>KontsevichIntegral[R,Phi,Knot[3,1]] // ReduceFI // CD </nowiki>		{{InOut\|n=13\|in=<nowiki>KontsevichIntegral[R,Phi,Knot[3,1]] // ReduceFI // CD </nowiki>
	\|out = <nowiki>1 + (23*CD[Circle[1, 2, 1, 2]])/24 -		\|out = <nowiki>1 + (23*CD[Circle[1, 2, 1, 2]])/24 -
	CD[Circle[1, 2, 3, 1, 2, 3]]/2 + (119*CD[Circle[1, 2, 3, 1, 4, 2, 3, 4]])/360 -		CD[Circle[1, 2, 3, 1, 2, 3]]/2 + (119*CD[Circle[1, 2, 3, 1, 4, 2, 3, 4]])/360 -
	(119CD[Circle[1, 2, 3, 1, 4, 3, 2, 4]])/720 + (247CD[Circle[1, 2, 3, 4, 1, 2, 3, 4]])/5760 </nowiki>}}		(119CD[Circle[1, 2, 3, 1, 4, 3, 2, 4]])/720 + (247CD[Circle[1, 2, 3, 4, 1, 2, 3, 4]])/5760 </nowiki>}}
	Below is another embedding of the trefoil, and the Kontsevich integral computed on this diagram is the same as that of the first:		Below is another embedding of the trefoil, and the Kontsevich integral computed on this diagram is the same as that of the first:
			[[Image:Trefoil.jpg\|left\|thumb\|80px\|The trefoil]]
	{{In\|n=15\|in=<nowiki>trefoil = MorseLink[Cup[1, 2], Cup[2, 3], X[1, Under, Down, Down],		{{In\|n=14\|in=<nowiki>trefoil = MorseLink[Cup[1, 2], Cup[2, 3], X[1, Under, Down, Down],
	X[3, Under, Up, Up], X[2, Over, Down, Up], Cap[4, 3], Cap[2, 1]];</nowiki>}}		X[3, Under, Up, Up], X[2, Over, Down, Up], Cap[4, 3], Cap[2, 1]];</nowiki>}}
	{{InOut\|n=16\|in=<nowiki>KontsevichIntegral[R,Phi,trefoil] // ReduceFI // CD</nowiki>		{{InOut\|n=16\|in=<nowiki>KontsevichIntegral[R,Phi,trefoil] // ReduceFI // CD</nowiki>
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	(119CD[Circle[1, 2, 3, 1, 4, 3, 2, 4]])/720 + (247CD[Circle[1, 2, 3, 4, 1, 2, 3, 4]])/5760 </nowiki>}}		(119CD[Circle[1, 2, 3, 1, 4, 3, 2, 4]])/720 + (247CD[Circle[1, 2, 3, 4, 1, 2, 3, 4]])/5760 </nowiki>}}

			<br style="clear:both" />
	==Performance Issues==		==Performance Issues==

Sankaran at 00:37, 13 September 2006

2006-09-13T00:37:31Z

← Older revision		Revision as of 20:37, 12 September 2006
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	In these examples, we shall continue to use the pair <math>(R, \Phi) </math> defined in the previous section.		In these examples, we shall continue to use the pair <math>(R, \Phi) </math> defined in the previous section.

		⚫	Here is the Kontsevich integral, up to degree four, of the trefoil:
	First, let's make sure we get the same answer as the previous section with this presentation of the unknot:
		⚫	{{InOut\|n=14\|in=<nowiki>KontsevichIntegral[R,Phi,Knot[3,1]] // ReduceFI // CD </nowiki>
	[[Image:Unknot.jpg\|thumb\|60px\|left\|The unknot]]
	~~{{In~~\|~~n=14\|in~~=<nowiki>~~unknot~~ = ~~MorseLink~~[~~Cup~~[1, 2], ~~Cup[3~~, 4], ~~X[2, Over, Up, Down],~~		\|out = <nowiki>1 + (23*CD[Circle[1, 2, 1, 2]])/24 -
	X[2,~~Under~~, ~~Down~~, ~~Up]~~,~~Cap[~~2, 3], ~~Cap~~[2, 1]];</~~nowiki>}}~~		CD[Circle[1, 2, 3, 1, 2, 3]]/2 + (119*CD[Circle[1, 2, 3, 1, 4, 2, 3, 4]])/360 -
			(119CD[Circle[1, 2, 3, 1, 4, 3, 2, 4]])/720 + (247CD[Circle[1, 2, 3, 4, 1, 2, 3, 4]])/5760 </nowiki>}}
⚫	~~We'll~~ ~~compute~~ the Kontsevich integral, ~~reduce~~ ~~it,~~ ~~and~~ ~~print~~ it ~~out~~:
			Below is another embedding of the trefoil, and the Kontsevich integral computed on this diagram is the same as that of the first:
⚫	{{InOut\|n=15\|in=<nowiki>KontsevichIntegral[R, Phi, ~~unknot~~] // ReduceFI // CD </nowiki>
	\|~~out~~ = <nowiki>1 - CD[~~Circle~~[1, 2, ~~1, 2]]/24 - CD~~[~~Circle[1,~~ 2, 3, 1, 4, 2, 3, ~~4]]/360 +~~		{{In\|n=15\|in=<nowiki>trefoil = MorseLink[Cup[1, 2], Cup[2, 3], X[1, Under, Down, Down],
	~~CD[Circle[1,~~ 2, 3, 1, 4, 3, 2, ~~4]]/720 + (7*CD[Circle[1~~, 2, 3, 4, 1, 2, ~~3, 4~~]]~~)/5760~~ </nowiki> }}		X[3, Under, Up, Up], X[2, Over, Down, Up], Cap[4, 3], Cap[2, 1]];</nowiki>}}
			{{InOut\|n=16\|in=<nowiki>KontsevichIntegral[R,Phi,trefoil] // ReduceFI // CD</nowiki>

			\|out = <nowiki>1 + (23*CD[Circle[1, 2, 1, 2]])/24 -
			CD[Circle[1, 2, 3, 1, 2, 3]]/2 + (119*CD[Circle[1, 2, 3, 1, 4, 2, 3, 4]])/360 -
			(119CD[Circle[1, 2, 3, 1, 4, 3, 2, 4]])/720 + (247CD[Circle[1, 2, 3, 4, 1, 2, 3, 4]])/5760 </nowiki>}}

	==Performance Issues==		==Performance Issues==

Sankaran at 19:30, 9 September 2006

2006-09-09T19:30:22Z

← Older revision		Revision as of 15:30, 9 September 2006
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	First, let's make sure we get the same answer as the previous section with this presentation of the unknot:		First, let's make sure we get the same answer as the previous section with this presentation of the unknot:
	[[Image:Unknot.jpg\|thumb\|60px\|left\|The unknot]]		[[Image:Unknot.jpg\|thumb\|60px\|left\|The unknot]]
	{{In\|n=14\|in=<nowiki>unknot = MorseLink[Cup[1, 2], Cup[3, 4], X[2, ~~Under~~, Up, Down],		{{In\|n=14\|in=<nowiki>unknot = MorseLink[Cup[1, 2], Cup[3, 4], X[2, Over, Up, Down],
	X[2,~~Over~~, Down, Up],Cap[2, 3], Cap[2, 1]];</nowiki>}}		X[2,Under, Down, Up],Cap[2, 3], Cap[2, 1]];</nowiki>}}
	We'll compute the Kontsevich integral, reduce it, and print it out:		We'll compute the Kontsevich integral, reduce it, and print it out:
	{{InOut\|n=15\|in=<nowiki>KontsevichIntegral[R, Phi, unknot] // ReduceFI // CD </nowiki>		{{InOut\|n=15\|in=<nowiki>KontsevichIntegral[R, Phi, unknot] // ReduceFI // CD </nowiki>

Sankaran at 19:16, 9 September 2006

2006-09-09T19:16:19Z

← Older revision		Revision as of 15:16, 9 September 2006
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	In these examples, we shall continue to use the pair <math>(R, \Phi) </math> defined in the previous section.		In these examples, we shall continue to use the pair <math>(R, \Phi) </math> defined in the previous section.

	<small>'''Note (9/5/2006):'''' There are a few bugs in the program right now: first, one must have at least one crossing, and second, one cannot have a crossing when there exactly two strands. The latter is not a huge problem because of Reidemeister 1, but be forewarned. This will be fixed as soon as we get the program to accept links as well as knots. </small>

	First, let's make sure we get the same answer as the previous section with this presentation of the unknot:		First, let's make sure we get the same answer as the previous section with this presentation of the unknot:

Sankaran at 14:57, 6 September 2006

2006-09-06T14:57:57Z

← Older revision		Revision as of 10:57, 6 September 2006
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	CD[Line[1, 2, 4], Line[3], Line[1, 2, 3, 4]]/480 -		CD[Line[1, 2, 4], Line[3], Line[1, 2, 3, 4]]/480 -
	CD[Line[1, 3, 4], Line[2], Line[1, 2, 3, 4]]/480 +		CD[Line[1, 3, 4], Line[2], Line[1, 2, 3, 4]]/480 +
	CD[Line[2, 3, 4], Line[1], Line[1, 2, 3, 4]]/1440,		CD[Line[2, 3, 4], Line[1], Line[1, 2, 3, 4]]/1440
	]; </nowiki>}}		]; </nowiki>}}

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	1 + CD[Line[1], Line[1]]/2 + CD[Line[1, 2], Line[1, 2]]/8 +		1 + CD[Line[1], Line[1]]/2 + CD[Line[1, 2], Line[1, 2]]/8 +
	CD[Line[1, 2, 3], Line[1, 2, 3]]/48 +		CD[Line[1, 2, 3], Line[1, 2, 3]]/48 +
	CD[Line[1, 2, 3, 4], Line[1, 2, 3, 4]]/384,		CD[Line[1, 2, 3, 4], Line[1, 2, 3, 4]]/384
	];</nowiki>}}		];</nowiki>}}

Sankaran at 14:56, 6 September 2006

2006-09-06T14:56:23Z

Sankaran at 01:31, 6 September 2006

2006-09-06T01:31:43Z

← Older revision		Revision as of 21:31, 5 September 2006
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	First, let's make sure we get the same answer as the previous section with this presentation of the unknot:		First, let's make sure we get the same answer as the previous section with this presentation of the unknot:
	[[Image:Unknot.jpg\|thumb\|60px\|left\|The unknot]]		[[Image:Unknot.jpg\|thumb\|60px\|left\|The unknot]]
	{{In\|n=14\|in=<nowiki>unknot = MorseLink[Cup[1, 2], Cup[3, 4], X[2, ~~Over~~, Up, Down],		{{In\|n=14\|in=<nowiki>unknot = MorseLink[Cup[1, 2], Cup[3, 4], X[2, Under, Up, Down],
	X[2,~~Under~~, Down, Up],Cap[2, 3], Cap[2, 1]];</nowiki>}}		X[2,Over, Down, Up],Cap[2, 3], Cap[2, 1]];</nowiki>}}
	We'll compute the Kontsevich integral, reduce it, and print it out:		We'll compute the Kontsevich integral, reduce it, and print it out:
	{{InOut\|n=15\|in=<nowiki>KontsevichIntegral[R, Phi, unknot] // ReduceFI // CD </nowiki>		{{InOut\|n=15\|in=<nowiki>KontsevichIntegral[R, Phi, unknot] // ReduceFI // CD </nowiki>

Sankaran at 01:25, 6 September 2006

2006-09-06T01:25:04Z

← Older revision		Revision as of 21:25, 5 September 2006
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	<small>'''Note (9/5/2006):'''' There are a few bugs in the program right now: first, one must have at least one crossing, and second, one cannot have a crossing when there exactly two strands. The latter is not a huge problem because of Reidemeister 1, but be forewarned. This will be fixed as soon as we get the program to accept links as well as knots. </small>		<small>'''Note (9/5/2006):'''' There are a few bugs in the program right now: first, one must have at least one crossing, and second, one cannot have a crossing when there exactly two strands. The latter is not a huge problem because of Reidemeister 1, but be forewarned. This will be fixed as soon as we get the program to accept links as well as knots. </small>

	First, let's ~~try~~ this presentation of the unknot:		First, let's make sure we get the same answer as the previous section with this presentation of the unknot:
			[[Image:Unknot.jpg\|thumb\|60px\|left\|The unknot]]
	{{In\|n=14\|in=<nowiki>unknot = MorseLink[Cup[1, 2], Cup[3, 4], X[2, Over, Up, Down], ~~X[2,Under, Down, Up],Cap[2, 3], Cap[2, 1]];</nowiki>}}~~		{{In\|n=14\|in=<nowiki>unknot = MorseLink[Cup[1, 2], Cup[3, 4], X[2, Over, Up, Down],
			X[2,Under, Down, Up],Cap[2, 3], Cap[2, 1]];</nowiki>}}
	We'll compute the Kontsevich integral, reduce it, and print it out:		We'll compute the Kontsevich integral, reduce it, and print it out:
	{{InOut\|n=15\|in=<nowiki>KontsevichIntegral[R, Phi, unknot] // ReduceFI // CD </nowiki>		{{InOut\|n=15\|in=<nowiki>KontsevichIntegral[R, Phi, unknot] // ReduceFI // CD </nowiki>
	\|out = <nowiki>1 - CD[Circle[1, 2, 1, 2]]/24 - CD[Circle[1, 2, 3, 1, 4, 2, 3, 4]]/360 +		\|out = <nowiki>1 - CD[Circle[1, 2, 1, 2]]/24 - CD[Circle[1, 2, 3, 1, 4, 2, 3, 4]]/360 +
	CD[Circle[1, 2, 3, 1, 4, 3, 2, 4]]/720 + (7*CD[Circle[1, 2, 3, 4, 1, 2, 3, 4]])/5760 </nowiki> }}		CD[Circle[1, 2, 3, 1, 4, 3, 2, 4]]/720 + (7*CD[Circle[1, 2, 3, 4, 1, 2, 3, 4]])/5760 </nowiki> }}
	Thankfully, this agrees with the previous section.

Sankaran at 01:10, 6 September 2006

2006-09-06T01:10:58Z

← Older revision		Revision as of 21:10, 5 September 2006
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	This page documents the ChordsMod4T component of the [[VasCalc]] project. This page is also under construction.		This page documents the ChordsMod4T component of the [[VasCalc]] project. This page is also under construction.

	'''Warning (08/19/06):''' The documentation below may be slightly out of date, as we are in the process of changing the <code>ASeries</code> notation. This should be resolved within a few days - please bear with us.
	==Overview==		==Overview==
	This package provides a Mathematica interface to work with the space generated by chord diagrams on a fixed skeleton of lines and circles, modulo the 4-T relation. See Dror Bar-Natan's [[Survey of Finite Type Invariants]] for further details.		This package provides a Mathematica interface to work with the space generated by chord diagrams on a fixed skeleton of lines and circles, modulo the 4-T relation. See Dror Bar-Natan's [[Survey of Finite Type Invariants]] for further details.
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	The software requirements of this package are Mathematica 4.1 and Java 1.4 (or greater, in both cases). However, note that default installations of Mathematica 4.1 and up include a Java runtime binary, and require no further configuration for this package to work.		The software requirements of this package are Mathematica 4.1 and Java 1.4 (or greater, in both cases). However, note that default installations of Mathematica 4.1 and up include a Java runtime binary, and require no further configuration for this package to work.

			In order to use the <code>KontsevichIntegral</code>, you must also install the ''KnotTheory`'' package, which is available from [http://katlas.math.toronto.edu/wiki/The_Mathematica_Package_KnotTheory%60 The Knot Atlas].

	While there are no specific hardware requirements, the computations involved are rather resource-intensive. Users of low-end machines will see a smaller range of parameters for which the computation is practicable. We'll have an expected performance profile once work on this package is done.		While there are no specific hardware requirements, the computations involved are rather resource-intensive. Users of low-end machines will see a smaller range of parameters for which the computation is practicable. We'll have an expected performance profile once work on this package is done.

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	==Usage==		==Usage==
	====Initialization====		====Initialization====
			If you wish to compute the <code>KontsevichIntegral</code> (and really, why else would you be here?), start by loading in the ''KnotTheory`'' package in a Mathematica session:
	In a Mathematica session, load the definitions by typing:
			{{InOut\|n=0\|in=<nowiki><<KnotTheory`</nowiki>\|
			out=<nowiki>Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
			Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.</nowiki>}}

			Also, load in the VasCalc definitions:

	{{In\|n=1\|in=<nowiki> <</path_to_install_folder/CDinterface.m </nowiki>}}		{{In\|n=1\|in=<nowiki> <</path_to_install_folder/CDinterface.m </nowiki>}}
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	To come: definition of Z[B[n]], demonstration of Reidemeister moves. For now, [[VasCalc Documentation - An example\| reidemeister.nb]].		To come: definition of Z[B[n]], demonstration of Reidemeister moves. For now, [[VasCalc Documentation - An example\| reidemeister.nb]].

	====~~(Unframed)~~ ~~Knot Invariants~~====		====The Unknot====
	To compute <math>Z(K)</math> for a knot, we need to extend the definition of <math>Z</math> to include "caps" and "cups". In addition, framing independence forces us to introduce the FI relation - hence all the formulae in this section take place in the "further reduced" space of chord diagrams. We proceed using the <math>Z_{\infty}</math> correction term in [http://www.math.toronto.edu/~drorbn/papers/nat/nat.pdf]. To begin, let us choose a pair <math>(R, \Phi)</math> which solve the pentagon and hexagon equations to fourth degree:		To compute <math>Z(K)</math> for any knot, including the unknot, we need to extend the definition of <math>Z</math> to include "caps" and "cups". In addition, framing independence forces us to introduce the FI relation - hence all the formulae in this section take place in the "further reduced" space of chord diagrams. We proceed using the <math>Z_{\infty}</math> correction term in [http://www.math.toronto.edu/~drorbn/papers/nat/nat.pdf]. To begin, let us choose a pair <math>(R, \Phi)</math> which solve the pentagon and hexagon equations to fourth degree:

	{{In\|n=9\|in=<nowiki>Phi = ASeries[1 + CD[Line[1], Line[2], Line[1, 2]]/24 -		{{In\|n=9\|in=<nowiki>Phi = ASeries[1 + CD[Line[1], Line[2], Line[1, 2]]/24 -
Line 164:		Line 170:

	The definition of <math>Z_\infty</math> involves going up the third strand, down the second, and up the first strand of <math>\Phi</math>:		The definition of <math>Z_\infty</math> involves going up the third strand, down the second, and up the first strand of <math>\Phi</math>:
	{{InOut\|n=11\|in=<nowiki>Zinf = ConnectStrand[~~ConnectStrand[ReverseStrand[Phi,~~ 2], 3, 2], 2, 1]</nowiki>		{{InOut\|n=11\|in=<nowiki>Zinf = ConnectStrand[2,1, ConnectStrand[3,2, ReverseStrand[2,Phi] ] ]</nowiki>
	\|out=<nowiki>ASeries[1, 0, {«JavaObject[vectorSpace.Coefficient]»,		\|out=<nowiki>ASeries[1, 0, {«JavaObject[vectorSpace.Coefficient]»,
	«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]»,		«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]»,
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	out=<nowiki>1 - CD[Circle[1, 2, 1, 2]]/24 - CD[Circle[1, 2, 3, 1, 4, 2, 3, 4]]/360 +		out=<nowiki>1 - CD[Circle[1, 2, 1, 2]]/24 - CD[Circle[1, 2, 3, 1, 4, 2, 3, 4]]/360 +
	CD[Circle[1, 2, 3, 1, 4, 3, 2, 4]]/720 + (7*CD[Circle[1, 2, 3, 4, 1, 2, 3, 4]])/5760 </nowiki>}}		CD[Circle[1, 2, 3, 1, 4, 3, 2, 4]]/720 + (7*CD[Circle[1, 2, 3, 4, 1, 2, 3, 4]])/5760 </nowiki>}}

			====The Kontsevich integral for unframed knots====

			We can find the Kontsevich integral of any knot <code>K</code>, through the command <code>KontsevichIntegral[R, Phi, K]</code>, where <code>R</code> and <code>Phi</code> are <code>ASeries</code> objects as above, and <code>K</code> is a [http://katlas.math.toronto.edu/wiki/MorseLink_Presentations Morse link] presentation of a knot (or alternatively any presentation that ''KnotTheory`'' knows how to convert to a Morse link presentation). The output, for a knot, is an <code>ASeries</code> object on one circle, whose degree is determined by the degrees of <code>R</code> and <code>Phi</code>.

			In these examples, we shall continue to use the pair <math>(R, \Phi) </math> defined in the previous section.

			<small>'''Note (9/5/2006):'''' There are a few bugs in the program right now: first, one must have at least one crossing, and second, one cannot have a crossing when there exactly two strands. The latter is not a huge problem because of Reidemeister 1, but be forewarned. This will be fixed as soon as we get the program to accept links as well as knots. </small>

			First, let's try this presentation of the unknot:
			{{In\|n=14\|in=<nowiki>unknot = MorseLink[Cup[1, 2], Cup[3, 4], X[2, Over, Up, Down], X[2,Under, Down, Up],Cap[2, 3], Cap[2, 1]];</nowiki>}}
			We'll compute the Kontsevich integral, reduce it, and print it out:
			{{InOut\|n=15\|in=<nowiki>KontsevichIntegral[R, Phi, unknot] // ReduceFI // CD </nowiki>
			\|out = <nowiki>1 - CD[Circle[1, 2, 1, 2]]/24 - CD[Circle[1, 2, 3, 1, 4, 2, 3, 4]]/360 +
			CD[Circle[1, 2, 3, 1, 4, 3, 2, 4]]/720 + (7*CD[Circle[1, 2, 3, 4, 1, 2, 3, 4]])/5760 </nowiki> }}
			Thankfully, this agrees with the previous section.


	==Performance Issues==		==Performance Issues==

Sankaran: moved "Special ASeries" content to body

2006-08-19T21:51:51Z

moved "Special ASeries" content to body

← Older revision		Revision as of 17:51, 19 August 2006
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	Finally, to convert an <code>ASeries</code> back to a human-readable form, use the command: <code>CD[ AS ]</code>, where <code>AS</code> is the object to be converted.		Finally, to convert an <code>ASeries</code> back to a human-readable form, use the command: <code>CD[ AS ]</code>, where <code>AS</code> is the object to be converted.


	====Special Form of ASeries====

	When calculating the invariant of a knot or a braid, we multiply the invariant of
	the different "events" in the knot's presentation. In the case where this event
	is a crossing, one must keep in mind that two strands have been permuted. For
	this reason one must apply <code>PermuteStrand[]</code> in order to get the
	correct answer. This causes the resulting expressions to be rather inelegant.
	For this reason we have devised a new form of <code>ASeries[]</code> that keeps
	track of the permutation of the strands so one does not need to manually permute
	them when multiplying.

	This format only applies when there are no circles in our ASeries, as in the
	case of knots and braids. One can then define an ASeries using the format:
	<code>ASeries[Permutation_list, CV_list]</code>. Here, <code>Permutation</code> is a list
	of integers from <math>1</math> to </math>n</math>, where <math>n</math> is
	the number of lines in the diagram. <code>CV</code> is a list containing Chord vectors
	of increasing degree.

	====Checking the pentagon and hexagon relations====		====Checking the pentagon and hexagon relations====

← Older revision		Revision as of 10:56, 6 September 2006
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	To compute <math>Z(K)</math> for any knot, including the unknot, we need to extend the definition of <math>Z</math> to include "caps" and "cups". In addition, framing independence forces us to introduce the FI relation - hence all the formulae in this section take place in the "further reduced" space of chord diagrams. We proceed using the <math>Z_{\infty}</math> correction term in [http://www.math.toronto.edu/~drorbn/papers/nat/nat.pdf]. To begin, let us choose a pair <math>(R, \Phi)</math> which solve the pentagon and hexagon equations to fourth degree:		To compute <math>Z(K)</math> for any knot, including the unknot, we need to extend the definition of <math>Z</math> to include "caps" and "cups". In addition, framing independence forces us to introduce the FI relation - hence all the formulae in this section take place in the "further reduced" space of chord diagrams. We proceed using the <math>Z_{\infty}</math> correction term in [http://www.math.toronto.edu/~drorbn/papers/nat/nat.pdf]. To begin, let us choose a pair <math>(R, \Phi)</math> which solve the pentagon and hexagon equations to fourth degree:

	{{In\|n=9\|in=<nowiki>Phi = ASeries[~~1 + CD~~[~~Line[1~~], ~~Line[2]~~, ~~Line[1, 2]]/24 -~~		{{In\|n=9\|in=<nowiki>Phi = ASeries[Id[3],0,
			1 + CD[Line[1], Line[2], Line[1, 2]]/24 -
	(7*CD[Line[1], Line[2, 3, 4], Line[1, 2, 3, 4]])/5760 -		(7*CD[Line[1], Line[2, 3, 4], Line[1, 2, 3, 4]])/5760 -
	CD[Line[2], Line[1], Line[1, 2]]/24 +		CD[Line[2], Line[1], Line[1, 2]]/24 +
Line 162:		Line 163:
	CD[Line[1, 3, 4], Line[2], Line[1, 2, 3, 4]]/480 +		CD[Line[1, 3, 4], Line[2], Line[1, 2, 3, 4]]/480 +
	CD[Line[2, 3, 4], Line[1], Line[1, 2, 3, 4]]/1440,		CD[Line[2, 3, 4], Line[1], Line[1, 2, 3, 4]]/1440,
	~~3,0~~]; </nowiki>}}		]; </nowiki>}}

	{{In\|n=10\|in=<nowiki>R = ASeries[~~1 + CD[Line[1]~~, ~~Line[1]]/2 + CD[Line[~~1, 2], ~~Line[1, 2]]/8 +~~		{{In\|n=10\|in=<nowiki>R = ASeries[{2,1},0,
			1 + CD[Line[1], Line[1]]/2 + CD[Line[1, 2], Line[1, 2]]/8 +
	CD[Line[1, 2, 3], Line[1, 2, 3]]/48 +		CD[Line[1, 2, 3], Line[1, 2, 3]]/48 +
	CD[Line[1, 2, 3, 4], Line[1, 2, 3, 4]]/384,		CD[Line[1, 2, 3, 4], Line[1, 2, 3, 4]]/384,
	~~2, 0~~];</nowiki>}}		];</nowiki>}}

	The definition of <math>Z_\infty</math> involves going up the third strand, down the second, and up the first strand of <math>\Phi</math>:		The definition of <math>Z_\infty</math> involves going up the third strand, down the second, and up the first strand of <math>\Phi</math>:
	{{InOut\|n=11\|in=<nowiki>Zinf = ConnectStrand[2,1, ConnectStrand[3,2, ReverseStrand[2,Phi] ] ]</nowiki>		{{InOut\|n=11\|in=<nowiki>Zinf = ConnectStrand[2,1, ConnectStrand[3,2, ReverseStrand[2,Phi] ] ]</nowiki>
	\|out=<nowiki>ASeries[1, 0, {«JavaObject[vectorSpace.Coefficient]»,		\|out=<nowiki>ASeries[{1}, 0, {«JavaObject[vectorSpace.Coefficient]»,
	«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]»,		«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]»,
	«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]»}]</nowiki>}}		«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]»}]</nowiki>}}

	The invariant of the unknot <math>Z(O)</math> is given by closing <math>Z_\infty</math> to a circle (the first line below), and taking the inverse:		The invariant of the unknot <math>Z(O)</math> is given by closing <math>Z_\infty</math> to a circle (the first line below), and taking the inverse:
	{{InOut\|n=12\|in=<nowiki>ZinfInverseCircle = ASeries[CD[(Zinf)^(-1)] /. Line -> Circle~~, 0 , 1~~]</nowiki>\|		{{InOut\|n=12\|in=<nowiki>ZinfInverseCircle = ASeries[0,1,CD[(Zinf)^(-1)] /. Line -> Circle]</nowiki>\|
	out = <nowiki>ASeries[0, 1, {«JavaObject[vectorSpace.Coefficient]»,		out = <nowiki>ASeries[0, 1, {«JavaObject[vectorSpace.Coefficient]»,
	«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]»,		«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]»,