R-Matrix Invariants - Revision history

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

2009-05-28T03:17:08Z

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

← Older revision		Revision as of 23:17, 27 May 2009
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RacroCdelt at 11:25, 22 May 2009

2009-05-22T11:25:18Z

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Scott at 23:07, 28 May 2007

2007-05-28T23:07:45Z

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			===See also===
			See also [[QuantumGroups`]], a Mathematica package by Scott Morrison, now included in KnotTheory`, which computes arbitrary quantum knot invariants. (For now, it only uses braid representations, not morse link representations.)

Siddarth at 00:23, 21 September 2005

2005-09-21T00:23:49Z

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	In more detail, let <math>V</math> be a free module of finite rank over a commutative ring, and let <math> R, \bar{R} \in End(V \otimes V) </math> be invertible. Further, let there be invertible endomorphisms <math>M_\leftarrow, M_\rightarrow, M^\leftarrow, M^\rightarrow \in End(V) </math>; these correspond to McupL, McupR, McapL, and McapR respectively. Fixing a basis of V, we can regard all of these endomorphisms as matrices.		In more detail, let <math>V</math> be a free module of finite rank over a commutative ring, and let <math> R, \bar{R} \in End(V \otimes V) </math> be invertible. Further, let there be invertible endomorphisms <math>M_\leftarrow, M_\rightarrow, M^\leftarrow, M^\rightarrow \in End(V) </math>; these correspond to McupL, McupR, McapL, and McapR respectively. Fixing a basis of V, we can regard all of these endomorphisms as matrices.

	The following notation is useful: for <math>n = dim(V)</math>, <math>R_{ab}^{cd}</math> refers to the element of '''''R''''' in row <math>n(c-1)+d</math> ~~row~~ and column <math>n(a-1) + b</math>.		The following notation is useful: for <math>n = dim(V)</math>, <math>R_{ab}^{cd}</math> refers to the element of '''''R''''' in row <math>n(c-1)+d</math> and column <math>n(a-1) + b</math>.

	For an oriented knot or link <math>K</math>, REngine returns the product <math> \tau(K) </math>, which is computed as follows:		For an oriented knot or link <math>K</math>, REngine returns the product <math> \tau(K) </math>, which is computed as follows:

Siddarth at 23:19, 8 September 2005

2005-09-08T23:19:51Z

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	:<math>M^\leftarrow = M^\rightarrow = M_\leftarrow = M_\rightarrow = \begin{pmatrix} 0&t^{-1/4} \\ t^{1/4}&0 \end{pmatrix} </math>		:<math>M^\leftarrow = M^\rightarrow = M_\leftarrow = M_\rightarrow = \begin{pmatrix} 0&t^{-1/4} \\ t^{1/4}&0 \end{pmatrix} </math>

	Putting these definitions into Mathematica, we run the REngine on the [[3_1\|trefoil]]. Note that the Jones polynomial is usually normalized so that the ~~value~~ at the unknot [[0_1]] is 1, hence the term on the denominator.		Putting these definitions into Mathematica, we run the REngine on the [[3_1\|trefoil]]. Note that the Jones polynomial is usually normalized so that the polynomial of the unknot [[0_1]] is 1, hence the term on the denominator.
	<!--$$r={{Sqrt[t], 0, 0, 0},{0, 0, t, 0},{0,t, Sqrt[t] - t^(3/2), 0}, {0,0,0,Sqrt[t]}};		<!--$$r={{Sqrt[t], 0, 0, 0},{0, 0, t, 0},{0,t, Sqrt[t] - t^(3/2), 0}, {0,0,0,Sqrt[t]}};
	rb = Inverse[r]; mcupl=mcapl=mcupr=mcapr={{0, t^(-1/4)},{t^(1/4), 0}};$$-->		rb = Inverse[r]; mcupl=mcapl=mcupr=mcapr={{0, t^(-1/4)},{t^(1/4), 0}};$$-->

Siddarth at 23:17, 8 September 2005

2005-09-08T23:17:54Z

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	{{InOut\|		{{InOut\|
	n = 4 \|		n = 4 \|
	in = <nowiki>jones = REngine[Knot[3,1], r, rb, mcupl, mcapl, mcupr, mcapr]		in = <nowiki>jones = REngine[Knot[3,1], r, rb, mcupl, mcapl, mcupr, mcapr] / REngine[Knot[0,1], r, rb, mcupl, mcapr, mcupr, mcapr] //Apart</nowiki> \|
		⚫	out= <nowiki> -4 -3 1
	/ REngine[Knot[0,1], r, rb, mcupl, mcapr, mcupr, mcapr]</nowiki> \|
			-t + t + -
⚫	out= <nowiki>~~$Failed</nowiki>}}~~
			t</nowiki>}}
	<!--END-->		<!--END-->

Siddarth at 23:17, 8 September 2005

2005-09-08T23:17:36Z

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	<!--END-->		<!--END-->

	<!--$$jones = REngine[Knot[3,1], r, rb, mcupl, mcapl, mcupr, mcapr]		<!--$$jones = REngine[Knot[3,1], r, rb, mcupl, mcapl, mcupr, mcapr] / REngine[Knot[0,1], r, rb, mcupl, mcapr, mcupr, mcapr] //Apart$$-->
	/ REngine[Knot[0,1], r, rb, mcupl, mcapr, mcupr, mcapr]$$-->
	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{InOut\|		{{InOut\|

Siddarth at 23:16, 8 September 2005

2005-09-08T23:16:21Z

← Older revision		Revision as of 19:16, 8 September 2005
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	{{InOut\|		{{InOut\|
	n = 4 \|		n = 4 \|
	in = <nowiki>jones = REngine[Knot[3,1], r, rb, mcupl, mcapl, mcupr, mcapr] ~~/ REngine[Knot[0,1], r, rb, mcupl, mcapr, mcupr, mcapr]</nowiki> \|~~		in = <nowiki>jones = REngine[Knot[3,1], r, rb, mcupl, mcapl, mcupr, mcapr]
			/ REngine[Knot[0,1], r, rb, mcupl, mcapr, mcupr, mcapr]</nowiki> \|
	out= <nowiki>$Failed</nowiki>}}		out= <nowiki>$Failed</nowiki>}}
	<!--END-->		<!--END-->

Siddarth at 23:16, 8 September 2005

2005-09-08T23:16:04Z

← Older revision		Revision as of 19:16, 8 September 2005
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	{{InOut\|		{{InOut\|
	n = 4 \|		n = 4 \|
	in = <nowiki>jones = REngine[Knot[3,1], r, rb, mcupl, mcapl, mcupr, mcapr]		in = <nowiki>jones = REngine[Knot[3,1], r, rb, mcupl, mcapl, mcupr, mcapr] / REngine[Knot[0,1], r, rb, mcupl, mcapr, mcupr, mcapr]</nowiki> \|
	/ REngine[Knot[0,1], r, rb, mcupl, mcapr, mcupr, mcapr]</nowiki> \|
	out= <nowiki>$Failed</nowiki>}}		out= <nowiki>$Failed</nowiki>}}
	<!--END-->		<!--END-->

Siddarth at 23:15, 8 September 2005

2005-09-08T23:15:25Z

← Older revision		Revision as of 19:15, 8 September 2005
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	<!--$$r={{Sqrt[t], 0, 0, 0},{0, 0, t, 0},{0,t, Sqrt[t] - t^(3/2), 0}, {0,0,0,Sqrt[t]}};		<!--$$r={{Sqrt[t], 0, 0, 0},{0, 0, t, 0},{0,t, Sqrt[t] - t^(3/2), 0}, {0,0,0,Sqrt[t]}};
	rb = Inverse[r]; mcupl=mcapl=mcupr=mcapr={{0, t^(-1/4)},{t^(1/4), 0}};$$-->		rb = Inverse[r]; mcupl=mcapl=mcupr=mcapr={{0, t^(-1/4)},{t^(1/4), 0}};$$-->
			<!--Robot Land, no human edits to "END"-->
			{{In\|
			n = 3 \|
			in = <nowiki>r={{Sqrt[t], 0, 0, 0},{0, 0, t, 0},{0,t, Sqrt[t] - t^(3/2), 0}, {0,0,0,Sqrt[t]}};
			rb = Inverse[r]; mcupl=mcapl=mcupr=mcapr={{0, t^(-1/4)},{t^(1/4), 0}};</nowiki>}}
	<!--END-->		<!--END-->

	<!--$$jones = REngine[Knot[3,1], r, rb, mcupl, mcapl, mcupr, mcapr]		<!--$$jones = REngine[Knot[3,1], r, rb, mcupl, mcapl, mcupr, mcapr]
	/ REngine[Knot[0,1], r, rb, mcupl, mcapr, mcupr, mcapr]$$-->		/ REngine[Knot[0,1], r, rb, mcupl, mcapr, mcupr, mcapr]$$-->
			<!--Robot Land, no human edits to "END"-->
			{{InOut\|
			n = 4 \|
			in = <nowiki>jones = REngine[Knot[3,1], r, rb, mcupl, mcapl, mcupr, mcapr]
			/ REngine[Knot[0,1], r, rb, mcupl, mcapr, mcupr, mcapr]</nowiki> \|
			out= <nowiki>$Failed</nowiki>}}
	<!--END-->		<!--END-->