VasCalc Documentation - An example - Revision history

Zak: /* The Reidemeister 3 Move */

2006-08-16T11:32:29Z

The Reidemeister 3 Move

← Older revision		Revision as of 07:32, 16 August 2006
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	«JavaObject[ChordVector]», «JavaObject[ChordVector]», «JavaObject[ChordVector]»}]</nowiki>}}		«JavaObject[ChordVector]», «JavaObject[ChordVector]», «JavaObject[ChordVector]»}]</nowiki>}}

		⚫	{{InOut\|n=9\|in=<nowiki>rr = Z[B[4], 6].Z[B[3], 6].Z[B[4], 6]</nowiki>\|

⚫	{{InOut\|n=9\|in=<nowiki>~~ll =~~ rr = Z[B[4], 6].Z[B[3], 6].Z[B[4], 6]</nowiki>\|
	out=<nowiki>ASeries[{1, 2, 5, 4, 3, 6}, {«JavaObject[vectorSpace.Coefficient]»,		out=<nowiki>ASeries[{1, 2, 5, 4, 3, 6}, {«JavaObject[vectorSpace.Coefficient]»,
	«JavaObject[ChordVector]», «JavaObject[ChordVector]», «JavaObject[ChordVector]»}]/nowiki>}}		«JavaObject[ChordVector]», «JavaObject[ChordVector]», «JavaObject[ChordVector]»}]</nowiki>}}

Zak: /* The Reidemeister 3 Move */

2006-08-16T11:27:55Z

The Reidemeister 3 Move

← Older revision		Revision as of 07:27, 16 August 2006
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	</math>		</math>

	We calculate the right and left hand sides separately.		We calculate the right and left hand sides separately for <math>k=3</math> and <math>n=6</math>.

	{{InOut\|n=8\|in=<nowiki>ll = Z[B[3], 6].Z[B[4], 6].Z[B[3], 6]</nowiki>\|		{{InOut\|n=8\|in=<nowiki>ll = Z[B[3], 6].Z[B[4], 6].Z[B[3], 6]</nowiki>\|

Zak: /* The Reidemeister 3 Move */

2006-08-16T11:26:44Z

The Reidemeister 3 Move

← Older revision		Revision as of 07:26, 16 August 2006
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	Now we need to load the definitions of <math>\Phi</math> and <math>R</math> as		Now we need to load the definitions of <math>\Phi</math> and <math>R</math> as
	defined in Dror Bar-Natan's paper [http://www.math.toronto.edu/~drorbn/papers/nat/nat.pdf ~~<i>Non-Associative Tangles</i>~~]:		defined in Dror Bar-Natan's paper on Non-Associative Tangles[http://www.math.toronto.edu/~drorbn/papers/nat/nat.pdf]:

	{{InOut\|n=3\|in=<nowiki>Phi = ASeries[1 + (1/24)*CD[Line[1],		{{InOut\|n=3\|in=<nowiki>Phi = ASeries[1 + (1/24)*CD[Line[1],

Zak: /* The Reidemeister 3 Move */

2006-08-16T11:24:46Z

The Reidemeister 3 Move

← Older revision		Revision as of 07:24, 16 August 2006
Line 13:		Line 13:

	Now we need to load the definitions of <math>\Phi</math> and <math>R</math> as		Now we need to load the definitions of <math>\Phi</math> and <math>R</math> as
	defined in Dror's paper on Non-Associative Tangles:		defined in Dror Bar-Natan's paper [http://www.math.toronto.edu/~drorbn/papers/nat/nat.pdf <i>Non-Associative Tangles</i>]:

	{{InOut\|n=3\|in=<nowiki>Phi = ASeries[1 + (1/24)*CD[Line[1],		{{InOut\|n=3\|in=<nowiki>Phi = ASeries[1 + (1/24)*CD[Line[1],
Line 76:		Line 76:

	Voila!		Voila!

			==References==
			*<nowiki>[1]</nowiki> D. Bar-Natan, [http://www.math.toronto.edu/~drorbn/papers/nat/nat.pdf <i>Non-Associative Tangles</i>]. Geometric Topology (proceedings of the Georgia International Topology Conference, W. H. Kazez ed.), 139-183, Amer. Math. Soc. and International Press, Providence, 1997.

Zak: /* The Reidemeister 3 Move */

2006-08-16T11:22:10Z

The Reidemeister 3 Move

← Older revision		Revision as of 07:22, 16 August 2006
Line 4:		Line 4:

	We want to use [[VasCalc]] to verify the third Reidemeister move. This is meant		We want to use [[VasCalc]] to verify the third Reidemeister move. This is meant
	as a small example of how to use the [[VasCalc]] package.		as a small example of how to use the VasCalc package.

	The first couple of steps are to load up VasCalc.		The first couple of steps are to load up VasCalc.
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	«JavaObject[ChordVector]», «JavaObject[ChordVector]»}] </nowiki>}}		«JavaObject[ChordVector]», «JavaObject[ChordVector]»}] </nowiki>}}

	Now we define the definition of Z(B(k),n). Where Z stands for our (pseudo) Vassiliev		Now we define the definition of <math>Z(B(k),n)</math>. Where <math>Z</math> stands for our (pseudo) Vassiliev invariant and <math>Z(B(k),n)</math> is the invariant of a braid on <math>n</math> strands with a crossing on the strands <math>k</math> and <math>k+1</math>.
	invariant and Z(B(k),n) is the invariant of a braid on n strands with a crossing
	on the strands k and k+1.


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	ASeries[tbl, ser[[3]]]]</nowiki>}}		ASeries[tbl, ser[[3]]]]</nowiki>}}

	Here we need to make an exception for ~~when~~ k is 1 because the above will not work:		Here we need to make an exception for the case <math>k=1</math> because the above will not work:

	{{In\|n=7\|in=<nowiki>Z[B[1], n_] := Module[{tbl, ser},		{{In\|n=7\|in=<nowiki>Z[B[1], n_] := Module[{tbl, ser},
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	ASeries[tbl, ser[[3]]]]</nowiki>}}		ASeries[tbl, ser[[3]]]]</nowiki>}}

	Now we are ready to verify the move. If Z is an invariant of the Reidemeister three move		Now we are ready to verify the move. If <math>Z</math> is an invariant of the Reidemeister three move
	we must have:		we must have:

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	{{InOut\|n=8\|in=<nowiki>ll = Z[B[3], 6].Z[B[4], 6].Z[B[3], 6]</nowiki>\|		{{InOut\|n=8\|in=<nowiki>ll = Z[B[3], 6].Z[B[4], 6].Z[B[3], 6]</nowiki>\|
	out=ASeries[{1, 2, 5, 4, 3, 6}, {«JavaObject[vectorSpace.Coefficient]»,		out=<nowiki>ASeries[{1, 2, 5, 4, 3, 6}, {«JavaObject[vectorSpace.Coefficient]»,
	«JavaObject[ChordVector]», «JavaObject[ChordVector]», «JavaObject[ChordVector]»}]</nowiki>}}		«JavaObject[ChordVector]», «JavaObject[ChordVector]», «JavaObject[ChordVector]»}]</nowiki>}}


	{{InOut\|n=9\|in=<nowiki>ll = rr = Z[B[4], 6].Z[B[3], 6].Z[B[4], 6]</nowiki>\|		{{InOut\|n=9\|in=<nowiki>ll = rr = Z[B[4], 6].Z[B[3], 6].Z[B[4], 6]</nowiki>\|
	out=ASeries[{1, 2, 5, 4, 3, 6}, {«JavaObject[vectorSpace.Coefficient]»,		out=<nowiki>ASeries[{1, 2, 5, 4, 3, 6}, {«JavaObject[vectorSpace.Coefficient]»,
	«JavaObject[ChordVector]», «JavaObject[ChordVector]», «JavaObject[ChordVector]»}]/nowiki>}}		«JavaObject[ChordVector]», «JavaObject[ChordVector]», «JavaObject[ChordVector]»}]/nowiki>}}

Zak: /* The Reidemeister 3 Move */

2006-08-16T11:16:48Z

The Reidemeister 3 Move

← Older revision		Revision as of 07:16, 16 August 2006
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	== The Reidemeister 3 Move ==		== The Reidemeister 3 Move ==

	We want to use VasCalc to verify the third Reidemeister move. This is meant		We want to use [[VasCalc]] to verify the third Reidemeister move. This is meant
	as a small example of how to use the VasCalc package.		as a small example of how to use the [[VasCalc]] package.

	The first couple of steps are to load up VasCalc.		The first couple of steps are to load up VasCalc.

Zak: /* The Reidemeister 3 Move */

2006-08-16T11:15:57Z

The Reidemeister 3 Move

@@ Line 39: / Line 39: @@
    Nest[AddStrand[#, #[[1]]] &, Nest[AddStrand[#, 0] &, R, k - 1], n - k - 1] .
    Nest[AddStrand[#, #[[1]]] &, Nest[DoubleStrand[#, 0] &,
-   PermuteStrand[((Phi))^(-1), {{2, 3}}], k - 2], n - k - 1];
+   PermuteStrand[(Phi)^(-1), {{2, 3}}], k - 2], n - k - 1];
   tbl = Table[i, {i, ser[[1]]}];
   tbl = ReplacePart[tbl, k + 1, k];
   tbl = ReplacePart[tbl, k, k + 1];
   ASeries[tbl, ser[[3]]]]</nowiki>}}

Zak: /* The Reidemeister 3 Move */

2006-08-16T11:06:37Z

The Reidemeister 3 Move

@@ Line 24: / Line 24: @@
 + (1/8)CD[Line[1, 2], Line[1, 2]] + (1/48)CD[Line[1, 2, 3], Line[1, 2, 3]]
 , 2, 0]</nowiki>|
-out=<nowiki>ASeries[2, 0, {«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]», «JavaObject[ChordVector]», «JavaObject[ChordVector]»}] </nowiki>}}
+out=<nowiki>ASeries[2, 0, {«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]»,

Zak: /* The Reidemeister 3 Move */

2006-08-16T10:52:46Z

The Reidemeister 3 Move

← Older revision		Revision as of 06:52, 16 August 2006
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	«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]»,		«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]»,
	«JavaObject[vectorSpace.Coefficient]»}] </nowiki>}}		«JavaObject[vectorSpace.Coefficient]»}] </nowiki>}}


			{{InOut\|n=4\|in=<nowiki>R = ASeries[1 + (1/2)CD[Line[1], Line[1]]
			+ (1/8)CD[Line[1, 2], Line[1, 2]] + (1/48)CD[Line[1, 2, 3], Line[1, 2, 3]]
			, 2, 0]</nowiki>\|
			out=<nowiki>ASeries[2, 0, {«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]», «JavaObject[ChordVector]», «JavaObject[ChordVector]»}] </nowiki>}}

Zak: /* The Reidemeister 3 Move */

2006-08-16T10:51:20Z

The Reidemeister 3 Move

← Older revision		Revision as of 06:51, 16 August 2006
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	{{In\|n=2\|in=<nowiki>SetVasCalcPath["/home/zavosh/vc"]; </nowiki>}}		{{In\|n=2\|in=<nowiki>SetVasCalcPath["/home/zavosh/vc"]; </nowiki>}}

			Now we need to load the definitions of <math>\Phi</math> and <math>R</math> as
			defined in Dror's paper on Non-Associative Tangles:

			{{InOut\|n=3\|in=<nowiki>Phi = ASeries[1 + (1/24)*CD[Line[1],
			Line[2], Line[1, 2]] - (1/24)*CD[Line[2], Line[1], Line[1, 2]], 3, 0, 3]</nowiki>\| out=<nowiki>ASeries[3, 0, {«JavaObject[vectorSpace.Coefficient]»,
			«JavaObject[vectorSpace.Coefficient]», «JavaObject[ChordVector]»,
			«JavaObject[vectorSpace.Coefficient]»}] </nowiki>}}