MorseLink Presentations - Revision history

Drorbn: /* Further notes */

2007-07-17T21:06:15Z

Further notes

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	{{HelpAndAbout\|		{{HelpAndAbout\|
	n = 1 \|		n = 2 \|
	n1 = 2 \|		n1 = 3 \|
	in = <nowiki>MorseLink</nowiki> \|		in = <nowiki>MorseLink</nowiki> \|
	out= <nowiki>MorseLink[K] returns a presentation of the oriented link K, composed, in successive order, of the following 'events': Cup[m,n] is a directed creation, starting at strand position n, towards position m, where m and n differ by 1. X[n,a = {Over/Under}, b = {Up/Down}, c={Up/Down}] is a crossing with lower-left edge at strand n, a determines whether the strand running bottom-left to top-right is over/under the crossing, b and c give the directions of the bottom-left and bottom-right strands respectively through the crossing. Cap[m,n] is a directed cap, from strand m to strand n.</nowiki> \|		out= <nowiki>MorseLink[K] returns a presentation of the oriented link K, composed, in successive order, of the following 'events': Cup[m,n] is a directed creation, starting at strand position n, towards position m, where m and n differ by 1. X[n,a = {Over/Under}, b = {Up/Down}, c={Up/Down}] is a crossing with lower-left edge at strand n, a determines whether the strand running bottom-left to top-right is over/under the crossing, b and c give the directions of the bottom-left and bottom-right strands respectively through the crossing. Cap[m,n] is a directed cap, from strand m to strand n.</nowiki> \|

Drorbn at 20:53, 17 July 2007

2007-07-17T20:53:10Z

← Older revision		Revision as of 16:53, 17 July 2007
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		⚫	[[Image:MLPres2.png\|250px\|left\|Two cap events]]After three more crossings, we encounter two cap events. The first annihilates strands 2 and 3, and is directed from 2 to 3. The second annihilates strands 1 and 2, but is directed from 2 to 1. So the MorseLink presentation for the knot [[4_1]] is given by: <code>MorseLink[Cup[1,2], Cup[3,4], X[2 , Under, Up, Down], X[2, Under, Down, Up], X[1, Over, Down, Up], X[1, Over, Up, Down], Cap[2,3], Cap[2,1]]</code>.

⚫	[[Image:MLPres2.png\|250px\|~~right~~\|Two cap events]]After three more crossings, we encounter two cap events. The first annihilates strands 2 and 3, and is directed from 2 to 3. The second annihilates strands 1 and 2, but is directed from 2 to 1. So the MorseLink presentation for the knot [[4_1]] is given by: <code>MorseLink[Cup[1,2], Cup[3,4], X[2 , Under, Up, Down], X[2, Under, Down, Up], X[1, Over, Down, Up], X[1, Over, Up, Down], Cap[2,3], Cap[2,1]]</code>.

	<br style="clear:both" />
	====Further notes====		====Further notes====

	*KnotTheory` knows about Morse link presentations:		KnotTheory` knows about Morse link presentations:
	{{Startup Note}}		{{Startup Note}}
	<!--$$?MorseLink$$-->		<!--$$?MorseLink$$-->

Drorbn at 20:47, 17 July 2007

2007-07-17T20:47:12Z

← Older revision		Revision as of 16:47, 17 July 2007
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	{{Manual TOC Sidebar}}		{{Manual TOC Sidebar}}

	The MorseLink presentation describes an oriented knot or link diagram as a sequence of 'events'. To begin, we fix an axis in the plane, so that each event in the MorseLink description occurs in successive intervals along this axis. The 'events' that comprise a knot or link are 'cups' (creations), 'caps' (annihilations), and crossings, where cups and caps are the minima and maxima (respectively) relative to the chosen axis. The conventions used by MorseLink are as follows:		The MorseLink presentation describes an oriented knot or link diagram as a sequence of 'events'. To begin, we fix an axis in the plane, so that each event in the MorseLink description occurs in successive intervals along this axis. The 'events' that comprise a knot or link are 'cups' (creations), 'caps' (annihilations), and crossings, where cups and caps are the minima and maxima (respectively) relative to the chosen axis. The conventions used by MorseLink are as follows:

Drorbn: Minor touchup.

2006-02-11T21:56:20Z

Minor touchup.

← Older revision		Revision as of 17:56, 11 February 2006
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	The MorseLink presentation describes an oriented knot or a link diagram as a sequence of 'events'. To begin, we fix an axis in the plane, so that each event in the MorseLink description occurs in successive intervals along this axis. The 'events' that comprise a knot or link are 'cups' (creations), 'caps' (annihilations), and crossings, where cups and caps are the minima and maxima (~~resp.~~) relative to the chosen axis. The conventions used by MorseLink are as follows:		The MorseLink presentation describes an oriented knot or link diagram as a sequence of 'events'. To begin, we fix an axis in the plane, so that each event in the MorseLink description occurs in successive intervals along this axis. The 'events' that comprise a knot or link are 'cups' (creations), 'caps' (annihilations), and crossings, where cups and caps are the minima and maxima (respectively) relative to the chosen axis. The conventions used by MorseLink are as follows:

	*'''Cup[m,n]''' is a creation, where the new strands are in position m and n, with m and n differing by 1, and is directed from m to n.		*'''Cup[m,n]''' represents a creation, where the new strands are in position m and n, with m and n differing by 1, and is directed from m to n.
	*'''Cap[m,n]''' is an annihilation of the strands m and n, again with m and n differing by 1, and is directed from m to n.		*'''Cap[m,n]''' represents an annihilation of the strands m and n, again with m and n differing by 1, and is directed from m to n.
	*'''X[n, d, a, b]''' is a crossing of the n-th and the (n+1)-th strands. 'd' signifies whether the n-th strand goes 'Over' or 'Under' the (n+1)-th strand. 'a' indicates whether the n-th strand is moving 'Up' or 'Down' through the crossing, relative to the chosen axis; 'b' does the same for the (n+1)-th strand.		*'''X[n, d, a, b]''' is a crossing of the n-th and the (n+1)-th strands. 'd' signifies whether the n-th strand goes 'Over' or 'Under' the (n+1)-th strand. 'a' indicates whether the n-th strand is moving 'Up' or 'Down' through the crossing, relative to the chosen axis; 'b' does the same for the (n+1)-th strand.

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	[[Image:MLPres0.png\|frame\|center\|A diagram of the knot [[4_1]] ]]		[[Image:MLPres0.png\|frame\|center\|A diagram of the knot [[4_1]] ]]

	We~~'ll~~ fix the chosen axis to be ~~horizontal~~ ~~and~~ to ~~the right~~, and we~~'ll~~ number the strands at each interval from bottom to top. Clearly the first event to take place is a cup, creating strands 1 and 2, and directed from 1 to 2. The next event is also a cup; this time, the strands 3 and 4 are created, from 3 to 4. So the first two elements of our MorseLink presentation are <code>{Cup[1,2], Cup[3,4], ...}</code>.		We fix the chosen axis to be the positive <math>x</math> axis, and we number the strands at each interval from bottom to top. Clearly the first event to take place is a cup, creating strands 1 and 2, and directed from 1 to 2. The next event is also a cup; this time, the strands 3 and 4 are created, from 3 to 4. So the first two elements of our MorseLink presentation are <code>{Cup[1,2], Cup[3,4], ...}</code>.

	[[Image:MLPres1.png\|250px\|left\|A crossing event]]The next event to take place is a crossing. This crossing has strand 2 going under strand 3; following the orientations, strand 2 moves to the right through the crossing, while strand 3 comes from the left. Since the chosen axis is pointing to the right, this event is described as <code>X[2, Under, Up, Down]</code>. We may proceed in a similar way for the rest of the crossings.		[[Image:MLPres1.png\|250px\|left\|A crossing event]]The next event to take place is a crossing. This crossing has strand 2 going under strand 3; following the orientations, strand 2 moves to the right through the crossing, while strand 3 comes from the left. Since the chosen axis is pointing to the right, this event is described as <code>X[2, Under, Up, Down]</code>. We may proceed in a similar way for the rest of the crossings.
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	[[Image:MLPres2.png\|250px\|right\|Two cap events]]After three more crossings, we encounter two cap events. The first annihilates strands 2 and 3, and is directed from 2 to 3. The second annihilates strands 1 and 2, but is directed from 2 to 1. So ~~our~~ MorseLink presentation for the knot [[4_1]] is given by: <code>MorseLink[Cup[1,2], Cup[3,4], X[2 , Under, Up, Down], X[2, Under, Down, Up], X[1, Over, Down, Up], X[1, Over, Up, Down], Cap[2,3], Cap[2,1]]</code>.		[[Image:MLPres2.png\|250px\|right\|Two cap events]]After three more crossings, we encounter two cap events. The first annihilates strands 2 and 3, and is directed from 2 to 3. The second annihilates strands 1 and 2, but is directed from 2 to 1. So the MorseLink presentation for the knot [[4_1]] is given by: <code>MorseLink[Cup[1,2], Cup[3,4], X[2 , Under, Up, Down], X[2, Under, Down, Up], X[1, Over, Down, Up], X[1, Over, Up, Down], Cap[2,3], Cap[2,1]]</code>.

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24.112.3.40 at 19:40, 2 September 2005

2005-09-02T19:40:03Z

← Older revision		Revision as of 15:40, 2 September 2005
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	<small> This page is under construction...</small>

	The MorseLink presentation describes an oriented knot or a link diagram as a sequence of 'events'. To begin, we fix an axis in the plane, so that each event in the MorseLink description occurs in successive intervals along this axis. The 'events' that comprise a knot or link are 'cups' (creations), 'caps' (annihilations), and crossings, where cups and caps are the minima and maxima (resp.) relative to the chosen axis. The conventions used by MorseLink are as follows:		The MorseLink presentation describes an oriented knot or a link diagram as a sequence of 'events'. To begin, we fix an axis in the plane, so that each event in the MorseLink description occurs in successive intervals along this axis. The 'events' that comprise a knot or link are 'cups' (creations), 'caps' (annihilations), and crossings, where cups and caps are the minima and maxima (resp.) relative to the chosen axis. The conventions used by MorseLink are as follows:

Siddarth at 18:48, 2 September 2005

2005-09-02T18:48:12Z

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	====Further notes====		====Further notes====

	*KnotTheory` knows about ~~MorseLink~~:		*KnotTheory` knows about Morse link presentations:
	{{Startup Note}}		{{Startup Note}}
	<!--$$?MorseLink$$-->		<!--$$?MorseLink$$-->

Siddarth at 18:46, 2 September 2005

2005-09-02T18:46:49Z

← Older revision		Revision as of 14:46, 2 September 2005
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	{{Startup Note}}		{{Startup Note}}
	<!--$$?MorseLink$$-->		<!--$$?MorseLink$$-->
			<!--Robot Land, no human edits to "END"-->
			{{HelpAndAbout\|
			n = 1 \|
			n1 = 2 \|
			in = <nowiki>MorseLink</nowiki> \|
			out= <nowiki>MorseLink[K] returns a presentation of the oriented link K, composed, in successive order, of the following 'events': Cup[m,n] is a directed creation, starting at strand position n, towards position m, where m and n differ by 1. X[n,a = {Over/Under}, b = {Up/Down}, c={Up/Down}] is a crossing with lower-left edge at strand n, a determines whether the strand running bottom-left to top-right is over/under the crossing, b and c give the directions of the bottom-left and bottom-right strands respectively through the crossing. Cap[m,n] is a directed cap, from strand m to strand n.</nowiki> \|
			about= <nowiki>MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005.</nowiki>}}
	<!--END-->		<!--END-->
	*Obviously, there is no unique Morse link presentation for a knot. One presentation may be considered 'better' than another if it generally has fewer strands. Unfortunately, this program makes only a half-hearted attempt in coming up with such presentations; it does a creation only when it can no longer do any caps or crossings, but exhibits a lack of foresight into which edges to create.		*Obviously, there is no unique Morse link presentation for a knot. One presentation may be considered 'better' than another if it generally has fewer strands. Unfortunately, this program makes only a half-hearted attempt in coming up with such presentations; it does a creation only when it can no longer do any caps or crossings, but exhibits a lack of foresight into which edges to create.

Siddarth at 18:40, 2 September 2005

2005-09-02T18:40:17Z

← Older revision		Revision as of 14:40, 2 September 2005
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	*KnotTheory` knows about MorseLink:		*KnotTheory` knows about MorseLink:
	{Startup Note}		{{Startup Note}}
	<!--$$?MorseLink$$-->		<!--$$?MorseLink$$-->
	<!--END-->		<!--END-->

Siddarth at 18:39, 2 September 2005

2005-09-02T18:39:24Z

← Older revision		Revision as of 14:39, 2 September 2005
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	We'll fix the chosen axis to be horizontal and to the right, and we'll number the strands at each interval from bottom to top. Clearly the first event to take place is a cup, creating strands 1 and 2, and directed from 1 to 2. The next event is also a cup; this time, the strands 3 and 4 are created, from 3 to 4. So the first two elements of our MorseLink presentation are <code>{Cup[1,2], Cup[3,4], ...}</code>.		We'll fix the chosen axis to be horizontal and to the right, and we'll number the strands at each interval from bottom to top. Clearly the first event to take place is a cup, creating strands 1 and 2, and directed from 1 to 2. The next event is also a cup; this time, the strands 3 and 4 are created, from 3 to 4. So the first two elements of our MorseLink presentation are <code>{Cup[1,2], Cup[3,4], ...}</code>.

	[[Image:MLPres1.png\|~~50px\|frame~~\|left\|A crossing event]]The next event to take place is a crossing. This crossing has strand 2 going under strand 3; following the orientations, strand 2 moves to the right through the crossing, while strand 3 comes from the left. Since the chosen axis is pointing to the right, this event is described as <code>X[2, Under, Up, Down]</code>. We may proceed in a similar way for the rest of the crossings.		[[Image:MLPres1.png\|250px\|left\|A crossing event]]The next event to take place is a crossing. This crossing has strand 2 going under strand 3; following the orientations, strand 2 moves to the right through the crossing, while strand 3 comes from the left. Since the chosen axis is pointing to the right, this event is described as <code>X[2, Under, Up, Down]</code>. We may proceed in a similar way for the rest of the crossings.

	<br style="clear:both" />


	[[Image:MLPres2.png\|~~frame~~\|~~left~~\|Two cap events]]After three more crossings, we encounter two cap events. The first annihilates strands 2 and 3, and is directed from 2 to 3. The second annihilates strands 1 and 2, but is directed from 2 to 1. So our MorseLink presentation for the knot [[4_1]] is given by: <code>MorseLink[Cup[1,2], Cup[3,4], X[2 , Under, Up, Down], X[2, Under, Down, Up], X[1, Over, Down, Up], X[1, Over, Up, Down], Cap[2,3], Cap[2,1]]</code>.		[[Image:MLPres2.png\|250px\|right\|Two cap events]]After three more crossings, we encounter two cap events. The first annihilates strands 2 and 3, and is directed from 2 to 3. The second annihilates strands 1 and 2, but is directed from 2 to 1. So our MorseLink presentation for the knot [[4_1]] is given by: <code>MorseLink[Cup[1,2], Cup[3,4], X[2 , Under, Up, Down], X[2, Under, Down, Up], X[1, Over, Down, Up], X[1, Over, Up, Down], Cap[2,3], Cap[2,1]]</code>.

	<br style="clear:both" />		<br style="clear:both" />
			====Further notes====

			*KnotTheory` knows about MorseLink:
			{Startup Note}
			<!--$$?MorseLink$$-->
			<!--END-->
			*Obviously, there is no unique Morse link presentation for a knot. One presentation may be considered 'better' than another if it generally has fewer strands. Unfortunately, this program makes only a half-hearted attempt in coming up with such presentations; it does a creation only when it can no longer do any caps or crossings, but exhibits a lack of foresight into which edges to create.

Siddarth at 18:25, 2 September 2005

2005-09-02T18:25:12Z

← Older revision		Revision as of 14:25, 2 September 2005
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	[[Image:MLPres0.png\|frame\|center\|A diagram of the knot [[4_1]] ]]		[[Image:MLPres0.png\|frame\|center\|A diagram of the knot [[4_1]] ]]

	We'll fix the chosen axis to be horizontal and to the right, and we'll number the strands at each interval from bottom to top.		We'll fix the chosen axis to be horizontal and to the right, and we'll number the strands at each interval from bottom to top. Clearly the first event to take place is a cup, creating strands 1 and 2, and directed from 1 to 2. The next event is also a cup; this time, the strands 3 and 4 are created, from 3 to 4. So the first two elements of our MorseLink presentation are <code>{Cup[1,2], Cup[3,4], ...}</code>.

			[[Image:MLPres1.png\|50px\|frame\|left\|A crossing event]]The next event to take place is a crossing. This crossing has strand 2 going under strand 3; following the orientations, strand 2 moves to the right through the crossing, while strand 3 comes from the left. Since the chosen axis is pointing to the right, this event is described as <code>X[2, Under, Up, Down]</code>. We may proceed in a similar way for the rest of the crossings.

			<br style="clear:both" />

			[[Image:MLPres2.png\|frame\|left\|Two cap events]]After three more crossings, we encounter two cap events. The first annihilates strands 2 and 3, and is directed from 2 to 3. The second annihilates strands 1 and 2, but is directed from 2 to 1. So our MorseLink presentation for the knot [[4_1]] is given by: <code>MorseLink[Cup[1,2], Cup[3,4], X[2 , Under, Up, Down], X[2, Under, Down, Up], X[1, Over, Down, Up], X[1, Over, Up, Down], Cap[2,3], Cap[2,1]]</code>.

			<br style="clear:both" />

← Older revision		Revision as of 17:06, 17 July 2007
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{HelpAndAbout\|		{{HelpAndAbout\|
	n = 1 \|		n = 2 \|
	n1 = 2 \|		n1 = 3 \|
	in = <nowiki>MorseLink</nowiki> \|		in = <nowiki>MorseLink</nowiki> \|
	out= <nowiki>MorseLink[K] returns a presentation of the oriented link K, composed, in successive order, of the following 'events': Cup[m,n] is a directed creation, starting at strand position n, towards position m, where m and n differ by 1. X[n,a = {Over/Under}, b = {Up/Down}, c={Up/Down}] is a crossing with lower-left edge at strand n, a determines whether the strand running bottom-left to top-right is over/under the crossing, b and c give the directions of the bottom-left and bottom-right strands respectively through the crossing. Cap[m,n] is a directed cap, from strand m to strand n.</nowiki> \|		out= <nowiki>MorseLink[K] returns a presentation of the oriented link K, composed, in successive order, of the following 'events': Cup[m,n] is a directed creation, starting at strand position n, towards position m, where m and n differ by 1. X[n,a = {Over/Under}, b = {Up/Down}, c={Up/Down}] is a crossing with lower-left edge at strand n, a determines whether the strand running bottom-left to top-right is over/under the crossing, b and c give the directions of the bottom-left and bottom-right strands respectively through the crossing. Cap[m,n] is a directed cap, from strand m to strand n.</nowiki> \|