Structure and Operations - Revision history

ScottDataRobot at 22:20, 21 February 2013

2013-02-21T22:20:33Z

← Older revision		Revision as of 18:20, 21 February 2013
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{HelpLine\|		{{HelpLine\|
	n = 1 \|		n = 2 \|
	in = <nowiki>Crossings</nowiki> \|		in = <nowiki>Crossings</nowiki> \|
	out= <nowiki>Crossings[L] returns the number of crossings of a knot/link L (in its given presentation).</nowiki>}}		out= <nowiki>Crossings[L] returns the number of crossings of a knot/link L (in its given presentation).</nowiki>}}
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{HelpLine\|		{{HelpLine\|
	n = 2 \|		n = 3 \|
	in = <nowiki>PositiveCrossings</nowiki> \|		in = <nowiki>PositiveCrossings</nowiki> \|
	out= <nowiki>PositiveCrossings[L] returns the number of positive (right handed) crossings in a knot/link L (in its given presentation).</nowiki>}}		out= <nowiki>PositiveCrossings[L] returns the number of positive (right handed) crossings in a knot/link L (in its given presentation).</nowiki>}}
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{HelpLine\|		{{HelpLine\|
	n = 3 \|		n = 4 \|
	in = <nowiki>NegativeCrossings</nowiki> \|		in = <nowiki>NegativeCrossings</nowiki> \|
	out= <nowiki>NegativeCrossings[L] returns the number of ~~negaitve~~ (left handed) crossings in a knot/link L (in its given presentation).</nowiki>}}		out= <nowiki>NegativeCrossings[L] returns the number of negative (left handed) crossings in a knot/link L (in its given presentation).</nowiki>}}
	<!--END-->		<!--END-->

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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{InOut\|		{{InOut\|
	n = 4 \|		n = 5 \|
	in = <nowiki>Crossings /@ {Knot[0, 1], TorusKnot[11,10]}</nowiki> \|		in = <nowiki>Crossings /@ {Knot[0, 1], TorusKnot[11,10]}</nowiki> \|
	out= <nowiki>{0, 99}</nowiki>}}		out= <nowiki>{0, 99}</nowiki>}}
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{InOut\|		{{InOut\|
	n = 5 \|		n = 6 \|
	in = <nowiki>K=Knot[6, 2]; {PositiveCrossings[K], NegativeCrossings[K]}</nowiki> \|		in = <nowiki>K=Knot[6, 2]; {PositiveCrossings[K], NegativeCrossings[K]}</nowiki> \|
	out= <nowiki>{2, 4}</nowiki>}}		out= <nowiki>{2, 4}</nowiki>}}
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{HelpLine\|		{{HelpLine\|
	n = 6 \|		n = 7 \|
	in = <nowiki>PositiveQ</nowiki> \|		in = <nowiki>PositiveQ</nowiki> \|
	out= <nowiki>PositiveQ[xing] returns True if xing is a positive (right handed) crossing and False if it is negative (left handed).</nowiki>}}		out= <nowiki>PositiveQ[xing] returns True if xing is a positive (right handed) crossing and False if it is negative (left handed).</nowiki>}}
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{HelpLine\|		{{HelpLine\|
	n = 7 \|		n = 8 \|
	in = <nowiki>NegativeQ</nowiki> \|		in = <nowiki>NegativeQ</nowiki> \|
	out= <nowiki>NegativeQ[xing] returns True if xing is a negative (left handed) crossing and False if it is positive (right handed).</nowiki>}}		out= <nowiki>NegativeQ[xing] returns True if xing is a negative (left handed) crossing and False if it is positive (right handed).</nowiki>}}
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{InOut\|		{{InOut\|
	n = 8 \|		n = 9 \|
	in = <nowiki>PositiveQ /@ {X[1,3,2,4], X[1,4,2,3], Xp[1,3,2,4], Xp[1,4,2,3]}</nowiki> \|		in = <nowiki>PositiveQ /@ {X[1,3,2,4], X[1,4,2,3], Xp[1,3,2,4], Xp[1,4,2,3]}</nowiki> \|
	out= <nowiki>{False, True, True, True}</nowiki>}}		out= <nowiki>{False, True, True, True}</nowiki>}}
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{HelpLine\|		{{HelpLine\|
	n = 9 \|		n = 10 \|
	in = <nowiki>ConnectedSum</nowiki> \|		in = <nowiki>ConnectedSum</nowiki> \|
	out= <nowiki>ConnectedSum[K1, K2] represents the connected sum of the knots K1 and K2 (ConnectedSum may not work with links).</nowiki>}}		out= <nowiki>ConnectedSum[K1, K2] represents the connected sum of the knots K1 and K2 (ConnectedSum may not work with links).</nowiki>}}
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{InOut\|		{{InOut\|
	n = 10 \|		n = 11 \|
	in = <nowiki>K = ConnectedSum[Knot[4,1], Knot[4,1]]</nowiki> \|		in = <nowiki>K = ConnectedSum[Knot[4,1], Knot[4,1]]</nowiki> \|
	out= <nowiki>ConnectedSum[Knot[4, 1], Knot[4, 1]]</nowiki>}}		out= <nowiki>ConnectedSum[Knot[4, 1], Knot[4, 1]]</nowiki>}}
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{InOut\|		{{InOut\|
	n = 11 \|		n = 12 \|
	in = <nowiki>Crossings[K]</nowiki> \|		in = <nowiki>Crossings[K]</nowiki> \|
	out= <nowiki>8</nowiki>}}		out= <nowiki>8</nowiki>}}
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{InOut\|		{{InOut\|
	n = 12 \|		n = 13 \|
	in = <nowiki>Jones[K][q] == Expand[Jones[Knot[4,1]][q]^2]</nowiki> \|		in = <nowiki>Jones[K][q] == Expand[Jones[Knot[4,1]][q]^2]</nowiki> \|
	out= <nowiki>True</nowiki>}}		out= <nowiki>True</nowiki>}}
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{InOut\|		{{InOut\|
	n = 13 \|		n = 14 \|
	in = <nowiki>Jones[K][q] == Jones[Knot[8,9]][q]</nowiki> \|		in = <nowiki>Jones[K][q] == Jones[Knot[8,9]][q]</nowiki> \|
	out= <nowiki>True</nowiki>}}		out= <nowiki>True</nowiki>}}
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{InOut\|		{{InOut\|
	n = 14 \|		n = 15 \|
	in = <nowiki>{Alexander[K][t], Alexander[Knot[8,9]][t]}</nowiki> \|		in = <nowiki>{Alexander[K][t], Alexander[Knot[8,9]][t]}</nowiki> \|
	out= <nowiki> -2 6 2 -3 3 5 2 3		out= <nowiki> -2 6 2 -3 3 5 2 3

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

2008-12-17T11:44:34Z

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

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

CooloCochi: deltabas

2008-12-16T17:04:13Z

deltabas

← Older revision		Revision as of 13:04, 16 December 2008
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			acrelvar
	{{Manual TOC Sidebar}}		{{Manual TOC Sidebar}}

Drorbn at 07:24, 3 September 2005

2005-09-03T07:24:49Z

← Older revision		Revision as of 03:24, 3 September 2005
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	It is also nice to know that, as expected, the Jones polynomial of <math>K</math> is the square of the Jones polynomial of [[4_1]]:		It is also nice to know that, as expected, the Jones polynomial of <math>K</math> is the square of the Jones polynomial of [[4_1]]:


	<!--$$Jones[K][q] == Expand[Jones[Knot[4,1]][q]^2]$$-->		<!--$$Jones[K][q] == Expand[Jones[Knot[4,1]][q]^2]$$-->
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	out= <nowiki>True</nowiki>}}		out= <nowiki>True</nowiki>}}
	<!--END-->		<!--END-->

			{{Knot Image Pair\|4_1\|gif\|8_9\|gif}}

	It is less nice to know that the Jones polynomial cannot tell <math>K</math> apart from the knot [[8_9]]:		It is less nice to know that the Jones polynomial cannot tell <math>K</math> apart from the knot [[8_9]]:

DrorsRobot at 17:11, 30 August 2005

2005-08-30T17:11:06Z

Show changes

69.156.15.171 at 00:43, 28 August 2005

2005-08-28T00:43:05Z

Show changes

Drorbn at 01:54, 25 August 2005

2005-08-25T01:54:23Z

← Older revision		Revision as of 21:54, 24 August 2005
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			{{Manual TOC Sidebar}}

	{{Startup Note}}		{{Startup Note}}

DrorsRobot at 01:52, 25 August 2005

2005-08-25T01:52:35Z

← Older revision		Revision as of 21:52, 24 August 2005
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	<!--$$?Crossings$$-->		<!--$$?Crossings$$-->
			<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
			{{Help1\|n=2\|s=Crossings}}
			Crossings[L] returns the number of crossings of a knot/link L (in its given presentation).
			{{Help2}}
	<!--END-->		<!--END-->

	<!--$$?PositiveCrossings$$-->		<!--$$?PositiveCrossings$$-->
			<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
			{{Help1\|n=3\|s=PositiveCrossings}}
			PositiveCrossings[L] returns the number of positive (right handed) crossings in a knot/link L (in its given presentation).
			{{Help2}}
	<!--END-->		<!--END-->

	<!--$$?NegativeCrossings$$-->		<!--$$?NegativeCrossings$$-->
			<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
			{{Help1\|n=4\|s=NegativeCrossings}}
			NegativeCrossings[L] returns the number of negaitve (left handed) crossings in a knot/link L (in its given presentation).
			{{Help2}}
	<!--END-->		<!--END-->

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	<!--$$Crossings /@ {Knot[0, 1], TorusKnot[11,10]}$$-->		<!--$$Crossings /@ {Knot[0, 1], TorusKnot[11,10]}$$-->
			<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
			{{InOut1\|n=5}}
			Crossings /@ {Knot[0, 1], TorusKnot[11,10]}
			{{InOut2\|n=5}}<pre style="border: 0px; padding: 0em"><nowiki>{0, 99}</nowiki></pre>
			{{InOut3}}
	<!--END-->		<!--END-->

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	<!--$$K=Knot[6, 2]; {PositiveCrossings[K], NegativeCrossings[K]}$$-->		<!--$$K=Knot[6, 2]; {PositiveCrossings[K], NegativeCrossings[K]}$$-->
			<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
			{{InOut1\|n=6}}
			K=Knot[6, 2]; {PositiveCrossings[K], NegativeCrossings[K]}
			{{InOut2\|n=6}}<pre style="border: 0px; padding: 0em"><nowiki>{2, 4}</nowiki></pre>
			{{InOut3}}
	<!--END-->		<!--END-->

	<!--$$?PositiveQ$$-->		<!--$$?PositiveQ$$-->
			<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
			{{Help1\|n=7\|s=PositiveQ}}
			PositiveQ[xing] returns True if xing is a positive (right handed) crossing and False if it is negative (left handed).
			{{Help2}}
	<!--END-->		<!--END-->

	<!--$$?NegativeQ$$-->		<!--$$?NegativeQ$$-->
			<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
			{{Help1\|n=8\|s=NegativeQ}}
			NegativeQ[xing] returns True if xing is a negative (left handed) crossing and False if it is positive (right handed).
			{{Help2}}
	<!--END-->		<!--END-->

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	<!--$$PositiveQ /@ {X[1,3,2,4], X[1,4,2,3], Xp[1,3,2,4], Xp[1,4,2,3]}$$-->		<!--$$PositiveQ /@ {X[1,3,2,4], X[1,4,2,3], Xp[1,3,2,4], Xp[1,4,2,3]}$$-->
			<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
			{{InOut1\|n=9}}
			PositiveQ /@ {X[1,3,2,4], X[1,4,2,3], Xp[1,3,2,4], Xp[1,4,2,3]}
			{{InOut2\|n=9}}<pre style="border: 0px; padding: 0em"><nowiki>{False, True, True, True}</nowiki></pre>
			{{InOut3}}
	<!--END-->		<!--END-->

	<!--$$?ConnectedSum$$-->		<!--$$?ConnectedSum$$-->
			<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
			{{Help1\|n=10\|s=ConnectedSum}}
			ConnectedSum[K1, K2] represents the connected sum of the knots K1 and K2 (ConnectedSum may not work with links).
			{{Help2}}
	<!--END-->		<!--END-->

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	<!--$$K = ConnectedSum[Knot[4,1], Knot[4,1]]$$-->		<!--$$K = ConnectedSum[Knot[4,1], Knot[4,1]]$$-->
			<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
			{{InOut1\|n=11}}
			K = ConnectedSum[Knot[4,1], Knot[4,1]]
			{{InOut2\|n=11}}<pre style="border: 0px; padding: 0em"><nowiki>ConnectedSum[Knot[4, 1], Knot[4, 1]]</nowiki></pre>
			{{InOut3}}
	<!--END-->		<!--END-->

	<!--$$Crossings[K]$$-->		<!--$$Crossings[K]$$-->
			<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
			{{InOut1\|n=12}}
			Crossings[K]
			{{InOut2\|n=12}}<pre style="border: 0px; padding: 0em"><nowiki>8</nowiki></pre>
			{{InOut3}}
	<!--END-->		<!--END-->

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	<!--$$Jones[K][q] == Expand[Jones[Knot[4,1]][q]^2]$$-->		<!--$$Jones[K][q] == Expand[Jones[Knot[4,1]][q]^2]$$-->
			<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
			{{InOut1\|n=13}}
			Jones[K][q] == Expand[Jones[Knot[4,1]][q]^2]
			{{InOut2\|n=13}}<pre style="border: 0px; padding: 0em"><nowiki>True</nowiki></pre>
			{{InOut3}}
	<!--END-->		<!--END-->

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	<!--$$Jones[K][q] == Jones[Knot[8,9]][q]$$-->		<!--$$Jones[K][q] == Jones[Knot[8,9]][q]$$-->
			<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
			{{InOut1\|n=14}}
			Jones[K][q] == Jones[Knot[8,9]][q]
			{{InOut2\|n=14}}<pre style="border: 0px; padding: 0em"><nowiki>True</nowiki></pre>
			{{InOut3}}
	<!--END-->		<!--END-->

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	<!--$${Alexander[K][t], Alexander[Knot[8,9]][t]}$$-->		<!--$${Alexander[K][t], Alexander[Knot[8,9]][t]}$$-->
			<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
			{{InOut1\|n=15}}
			{Alexander[K][t], Alexander[Knot[8,9]][t]}
			{{InOut2\|n=15}}<pre style="border: 0px; padding: 0em"><nowiki> -2 6 2 -3 3 5 2 3
			{11 + t - - - 6 t + t , 7 - t + -- - - - 5 t + 3 t - t }
			t 2 t
			t</nowiki></pre>
			{{InOut3}}
	<!--END-->		<!--END-->

Drorbn at 01:52, 25 August 2005

2005-08-25T01:52:25Z

← Older revision		Revision as of 21:52, 24 August 2005
Line 36:		Line 36:
	The connected sum <math>K=4_1\#4_1</math> of the knot [[4_1]] with itself has 8 crossings (unsurprisingly):		The connected sum <math>K=4_1\#4_1</math> of the knot [[4_1]] with itself has 8 crossings (unsurprisingly):

	<!--$$?K = ConnectedSum[Knot[4,1], Knot[4,1]]$$-->		<!--$$K = ConnectedSum[Knot[4,1], Knot[4,1]]$$-->
	<!--END-->		<!--END-->

Drorbn at 01:50, 25 August 2005

2005-08-25T01:50:14Z

← Older revision		Revision as of 21:50, 24 August 2005
Line 31:		Line 31:
	<!--END-->		<!--END-->

	<!--$$?ConnectedSum]$$-->		<!--$$?ConnectedSum$$-->
	<!--END-->		<!--END-->

	The connected sum <math>K=4_1\#4_1</math> of the knot [[4_1]] with itself has 8 crossings (unsurprisingly):		The connected sum <math>K=4_1\#4_1</math> of the knot [[4_1]] with itself has 8 crossings (unsurprisingly):

	<!--$$?K = ConnectedSum[Knot[4,1], Knot[4,1]$$-->		<!--$$?K = ConnectedSum[Knot[4,1], Knot[4,1]]$$-->
	<!--END-->		<!--END-->