Quantum knot invariants - Revision history

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

2009-05-28T03:13:37Z

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

← Older revision		Revision as of 23:13, 27 May 2009
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	http://www.textccaerl.com
	{{Manual TOC Sidebar}}		{{Manual TOC Sidebar}}

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	n1 = 2 \|		n1 = 2 \|
	in = <nowiki>QuantumKnotInvariant</nowiki> \|		in = <nowiki>QuantumKnotInvariant</nowiki> \|
	out= <nowiki>QuantumKnotInvariant[Î, V][K][q] calculates the quantum knot invariant of the knot K in the representation V of the quantum group Î. This relies on the QuantumGroups` package, and you should look there for details of how Î and V may be specified. Examples: QuantumKnotInvariant[Subscript[A,2], Irrep[Subscript[A,2]][{1, 0}]][Knot[5, 2]][q] QuantumKnotInvariant[Subscript[G,2], Irrep[Subscript[G,2]][{1, 0}]~~âIrrep~~[Subscript[G,2]][{0, 1}]][Knot[5, 2]][q]</nowiki> \|		out= <nowiki>QuantumKnotInvariant[Γ, V][K][q] calculates the quantum knot invariant of the knot K in the representation V of the quantum group Γ. This relies on the QuantumGroups` package, and you should look there for details of how Γ and V may be specified. Examples: QuantumKnotInvariant[Subscript[A,2], Irrep[Subscript[A,2]][{1, 0}]][Knot[5, 2]][q] QuantumKnotInvariant[Subscript[G,2], Irrep[Subscript[G,2]][{1, 0}]⊕Irrep[Subscript[G,2]][{0, 1}]][Knot[5, 2]][q]</nowiki> \|
	about= <nowiki>Quantum knot invariants are calculated using the mathematica package QuantumGroups`, written by Scott Morrison 2003-2006.</nowiki>}}		about= <nowiki>Quantum knot invariants are calculated using the mathematica package QuantumGroups`, written by Scott Morrison 2003-2006.</nowiki>}}
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DomleToboo at 11:10, 22 May 2009

2009-05-22T11:10:57Z

← Older revision		Revision as of 07:10, 22 May 2009
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			http://www.textccaerl.com
	{{Manual TOC Sidebar}}		{{Manual TOC Sidebar}}

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	n1 = 2 \|		n1 = 2 \|
	in = <nowiki>QuantumKnotInvariant</nowiki> \|		in = <nowiki>QuantumKnotInvariant</nowiki> \|
	out= <nowiki>QuantumKnotInvariant[Γ, V][K][q] calculates the quantum knot invariant of the knot K in the representation V of the quantum group Γ. This relies on the QuantumGroups` package, and you should look there for details of how Γ and V may be specified. Examples: QuantumKnotInvariant[Subscript[A,2], Irrep[Subscript[A,2]][{1, 0}]][Knot[5, 2]][q] QuantumKnotInvariant[Subscript[G,2], Irrep[Subscript[G,2]][{1, 0}]~~⊕Irrep~~[Subscript[G,2]][{0, 1}]][Knot[5, 2]][q]</nowiki> \|		out= <nowiki>QuantumKnotInvariant[Î, V][K][q] calculates the quantum knot invariant of the knot K in the representation V of the quantum group Î. This relies on the QuantumGroups` package, and you should look there for details of how Î and V may be specified. Examples: QuantumKnotInvariant[Subscript[A,2], Irrep[Subscript[A,2]][{1, 0}]][Knot[5, 2]][q] QuantumKnotInvariant[Subscript[G,2], Irrep[Subscript[G,2]][{1, 0}]âIrrep[Subscript[G,2]][{0, 1}]][Knot[5, 2]][q]</nowiki> \|
	about= <nowiki>Quantum knot invariants are calculated using the mathematica package QuantumGroups`, written by Scott Morrison 2003-2006.</nowiki>}}		about= <nowiki>Quantum knot invariants are calculated using the mathematica package QuantumGroups`, written by Scott Morrison 2003-2006.</nowiki>}}
	<!--END-->		<!--END-->

Scott at 17:47, 24 June 2006

2006-06-24T17:47:20Z

← Older revision		Revision as of 13:47, 24 June 2006
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	n1 = 2 \|		n1 = 2 \|
	in = <nowiki>QuantumKnotInvariant</nowiki> \|		in = <nowiki>QuantumKnotInvariant</nowiki> \|
	out= <nowiki>QuantumKnotInvariant[Γ, V][K][q] calculates the quantum knot invariant of the knot K in the representation V of the quantum group Γ. This relies on the QuantumGroups` package, and you should look there for details of how Γ and V may be specified. Examples: QuantumKnotInvariant[~~A2~~, Irrep[~~A2~~][{1, 0}]][Knot[5, 2]][q] QuantumKnotInvariant[~~G2~~, Irrep[~~G2~~][{1, 0}]⊕Irrep[~~G2~~][{0, 1}]][Knot[5, 2]][q]</nowiki> \|		out= <nowiki>QuantumKnotInvariant[Γ, V][K][q] calculates the quantum knot invariant of the knot K in the representation V of the quantum group Γ. This relies on the QuantumGroups` package, and you should look there for details of how Γ and V may be specified. Examples: QuantumKnotInvariant[Subscript[A,2], Irrep[Subscript[A,2]][{1, 0}]][Knot[5, 2]][q] QuantumKnotInvariant[Subscript[G,2], Irrep[Subscript[G,2]][{1, 0}]⊕Irrep[Subscript[G,2]][{0, 1}]][Knot[5, 2]][q]</nowiki> \|
	about= <nowiki>Quantum knot invariants are calculated using the mathematica package QuantumGroups`, written by Scott Morrison 2003-2006.</nowiki>}}		about= <nowiki>Quantum knot invariants are calculated using the mathematica package QuantumGroups`, written by Scott Morrison 2003-2006.</nowiki>}}
	<!--END-->		<!--END-->

Scott at 17:45, 24 June 2006

2006-06-24T17:45:15Z

@@ Line 89: / Line 89: @@
 <!--$$QuantumKnotInvariant[Subscript[D,4], Irrep[Subscript[D,4]][{0,1,0,0}]][Knot[5,1]][q]$$-->
 <!--END-->
 <!--$$qDimension[Subscript[B,2]][Irrep[Subscript[B,2]][{0,1}]] /. q->1$$-->
 <!--END-->
 <!--$$BR[Knot[7,4]]$$-->
 <!--END-->
 <!--$$QuantumKnotInvariant[Subscript[B,2], Irrep[Subscript[B,2]][{0,1}]][Knot[7,4]][q]$$-->
 <!--END-->

Scott at 17:44, 24 June 2006

2006-06-24T17:44:25Z

← Older revision		Revision as of 13:44, 24 June 2006
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	<!--END-->		<!--END-->

	<!--$$qDimension[Irrep[Subscript[B,3]][{0,1,0}]] /. q->1$$-->		<!--$$qDimension[Subscript[B,2]][Irrep[Subscript[B,2]][{0,1}]] /. q->1$$-->
	<!--END-->		<!--END-->
	<!--$$BR[Knot[7,4]]$$-->		<!--$$BR[Knot[7,4]]$$-->
	<!--END-->		<!--END-->
	<!--$$QuantumKnotInvariant[Subscript[B,3], Irrep[Subscript[B,3]][{0,1,0}]][Knot[7,4]][q]$$-->		<!--$$QuantumKnotInvariant[Subscript[B,2], Irrep[Subscript[B,2]][{0,1}]][Knot[7,4]][q]$$-->
	<!--END-->		<!--END-->

Scott at 17:42, 24 June 2006

2006-06-24T17:42:14Z

← Older revision		Revision as of 13:42, 24 June 2006
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	<!--END-->		<!--END-->

			<!--$$qDimension[Irrep[Subscript[B,3]][{0,1,0}]] /. q->1$$-->
			<!--END-->
			<!--$$BR[Knot[7,4]]$$-->
			<!--END-->
	<!--$$QuantumKnotInvariant[Subscript[B,3], Irrep[Subscript[B,3]][{0,1,0}]][Knot[7,4]][q]$$-->		<!--$$QuantumKnotInvariant[Subscript[B,3], Irrep[Subscript[B,3]][{0,1,0}]][Knot[7,4]][q]$$-->
	<!--END-->		<!--END-->

Scott at 17:41, 24 June 2006

2006-06-24T17:41:14Z

← Older revision		Revision as of 13:41, 24 June 2006
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	<!--END-->		<!--END-->

	<!--QuantumKnotInvariant[Subscript[B,3], Irrep[Subscript[B,3]][{0,1,0}]][Knot[7,4]][q]-->		<!--$$QuantumKnotInvariant[Subscript[B,3], Irrep[Subscript[B,3]][{0,1,0}]][Knot[7,4]][q]$$-->
	<!--~~E ND~~-->		<!--END-->

	The representations in the examples above are all irreducibles, specified by their highest weight. (The QuantumGroups` package represents weights by the coordinates in the fundamental basis. Thus <math>Irrep[A_3][{0,1,0}] = \wedge^2(C^4)</math>, for example. We can also specify other representations, as direct sums and tensor products.		The representations in the examples above are all irreducibles, specified by their highest weight. (The QuantumGroups` package represents weights by the coordinates in the fundamental basis. Thus <math>Irrep[A_3][{0,1,0}] = \wedge^2(C^4)</math>, for example. We can also specify other representations, as direct sums and tensor products.

			<!--Problems here...-->
	<!--$$QuantumKnotInvariant[Subscript[A,1], TensorProduct[DirectSum[Irrep[Subscript[A,1]][{1}], Irrep[Subscript[A,1]][{1}]],Irrep[Subscript[A,1]][{2}]]][Knot[5,1]][q]$$-->
		⚫	<!--(Tensor product and direct sum can be typeset much more prettily in a Mathematica notebook, using <code><esc>c*<esc></code> and <code><esc>c+<esc></code>.)-->
	<!--END-->

⚫	(Tensor product and direct sum can be typeset much more prettily in a Mathematica notebook, using <code><esc>c*<esc></code> and <code><esc>c+<esc></code>.)

Scott at 17:40, 24 June 2006

2006-06-24T17:40:05Z

← Older revision		Revision as of 13:40, 24 June 2006
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	<!--END-->		<!--END-->

	<!--QuantumKnotInvariant[Subscript[D,4], Irrep[Subscript[D,4]][{0,1,0,0}]][Knot[5,1]][q]-->		<!--$$QuantumKnotInvariant[Subscript[D,4], Irrep[Subscript[D,4]][{0,1,0,0}]][Knot[5,1]][q]$$-->
	<!--~~E ND~~-->		<!--END-->

	<!--QuantumKnotInvariant[Subscript[B,3], Irrep[Subscript[B,3]][{0,1,0}]][Knot[7,4]][q]-->		<!--QuantumKnotInvariant[Subscript[B,3], Irrep[Subscript[B,3]][{0,1,0}]][Knot[7,4]][q]-->
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	The representations in the examples above are all irreducibles, specified by their highest weight. (The QuantumGroups` package represents weights by the coordinates in the fundamental basis. Thus <math>Irrep[A_3][{0,1,0}] = \wedge^2(C^4)</math>, for example. We can also specify other representations, as direct sums and tensor products.		The representations in the examples above are all irreducibles, specified by their highest weight. (The QuantumGroups` package represents weights by the coordinates in the fundamental basis. Thus <math>Irrep[A_3][{0,1,0}] = \wedge^2(C^4)</math>, for example. We can also specify other representations, as direct sums and tensor products.

	<!--QuantumKnotInvariant[Subscript[A,1], TensorProduct[DirectSum[Irrep[Subscript[A,1]][{1}], Irrep[Subscript[A,1]][{1}]],Irrep[Subscript[A,1]][{2}]]][Knot[5,1]][q]-->		<!--$$QuantumKnotInvariant[Subscript[A,1], TensorProduct[DirectSum[Irrep[Subscript[A,1]][{1}], Irrep[Subscript[A,1]][{1}]],Irrep[Subscript[A,1]][{2}]]][Knot[5,1]][q]$$-->
	<!--~~E ND~~-->		<!--END-->

	(Tensor product and direct sum can be typeset much more prettily in a Mathematica notebook, using <code><esc>c*<esc></code> and <code><esc>c+<esc></code>.)		(Tensor product and direct sum can be typeset much more prettily in a Mathematica notebook, using <code><esc>c*<esc></code> and <code><esc>c+<esc></code>.)

Scott at 17:38, 24 June 2006

2006-06-24T17:38:05Z

@@ Line 9: / Line 9: @@
 <!--$$?QuantumKnotInvariant$$-->
 <!--END-->
@@ Line 14: / Line 21: @@
 <!--$${Jones[Knot[6,1]][q], QuantumKnotInvariant[Subscript[A,1], Irrep[Subscript[A,1]][{1}]][Knot[6,1]][q]}$$-->
 <!--END-->
 <!--$$Simplify[(q+q^(-1))Jones[Knot[6,1]][q^(-2)] - QuantumKnotInvariant[Subscript[A,1], Irrep[Subscript[A,1]][{1}]][Knot[6,1]][q]]$$-->
 <!--END-->
@@ Line 22: / Line 41: @@
 <!--$$QuantumKnotInvariant[Subscript[A,2], Irrep[Subscript[A,2]][{1,1}]][Knot[7,3]][q]$$-->
 <!--END-->
 <!--$$QuantumKnotInvariant[Subscript[A,3], Irrep[Subscript[A,3]][{0,1,0}]][Knot[8,2]][q]$$-->
 <!--END-->
 <!--$$QuantumKnotInvariant[Subscript[G,2], Irrep[Subscript[G,2]][{1,0}]][Knot[5,2]][q]$$-->
 <!--END-->

Scott at 17:37, 24 June 2006

2006-06-24T17:37:47Z

← Older revision		Revision as of 13:37, 24 June 2006
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	<!--END-->		<!--END-->

	<!--QuantumKnotInvariant[Subscript[D,4], Irrep[Subscript[D,4]][{0,1,0,0}]][Knot[5,1]][q]$$-->		<!--QuantumKnotInvariant[Subscript[D,4], Irrep[Subscript[D,4]][{0,1,0,0}]][Knot[5,1]][q]-->
	<!--~~END~~-->		<!--E ND-->

	<!--QuantumKnotInvariant[Subscript[B,3], Irrep[Subscript[B,3]][{0,1,0}]][Knot[7,4]][q]$$-->		<!--QuantumKnotInvariant[Subscript[B,3], Irrep[Subscript[B,3]][{0,1,0}]][Knot[7,4]][q]-->
	<!--~~END~~-->		<!--E ND-->

	The representations in the examples above are all irreducibles, specified by their highest weight. (The QuantumGroups` package represents weights by the coordinates in the fundamental basis. Thus <math>Irrep[A_3][{0,1,0}] = \wedge^2(C^4)</math>, for example. We can also specify other representations, as direct sums and tensor products.		The representations in the examples above are all irreducibles, specified by their highest weight. (The QuantumGroups` package represents weights by the coordinates in the fundamental basis. Thus <math>Irrep[A_3][{0,1,0}] = \wedge^2(C^4)</math>, for example. We can also specify other representations, as direct sums and tensor products.

	<!--QuantumKnotInvariant[Subscript[A,1], TensorProduct[DirectSum[Irrep[Subscript[A,1]][{1}], Irrep[Subscript[A,1]][{1}]],Irrep[Subscript[A,1]][{2}]]][Knot[5,1]][q]$$-->		<!--QuantumKnotInvariant[Subscript[A,1], TensorProduct[DirectSum[Irrep[Subscript[A,1]][{1}], Irrep[Subscript[A,1]][{1}]],Irrep[Subscript[A,1]][{2}]]][Knot[5,1]][q]-->
	<!--~~END~~-->		<!--E ND-->

	(Tensor product and direct sum can be typeset much more prettily in a Mathematica notebook, using <code><esc>c*<esc></code> and <code><esc>c+<esc></code>.)		(Tensor product and direct sum can be typeset much more prettily in a Mathematica notebook, using <code><esc>c*<esc></code> and <code><esc>c+<esc></code>.)