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	<id>https://katlas.org/index.php?action=history&amp;feed=atom&amp;title=Notes_for_3_1%27s_three_dimensional_invariants</id>
	<title>Notes for 3 1&#039;s three dimensional invariants - Revision history</title>
	<link rel="self" type="application/atom+xml" href="https://katlas.org/index.php?action=history&amp;feed=atom&amp;title=Notes_for_3_1%27s_three_dimensional_invariants"/>
	<link rel="alternate" type="text/html" href="https://katlas.org/index.php?title=Notes_for_3_1%27s_three_dimensional_invariants&amp;action=history"/>
	<updated>2026-07-11T14:03:43Z</updated>
	<subtitle>Revision history for this page on the wiki</subtitle>
	<generator>MediaWiki 1.39.6</generator>
	<entry>
		<id>https://katlas.org/index.php?title=Notes_for_3_1%27s_three_dimensional_invariants&amp;diff=1691719&amp;oldid=prev</id>
		<title>Drorbn at 18:15, 14 April 2008</title>
		<link rel="alternate" type="text/html" href="https://katlas.org/index.php?title=Notes_for_3_1%27s_three_dimensional_invariants&amp;diff=1691719&amp;oldid=prev"/>
		<updated>2008-04-14T18:15:59Z</updated>

		<summary type="html">&lt;p&gt;&lt;/p&gt;
&lt;table style=&quot;background-color: #fff; color: #202122;&quot; data-mw=&quot;interface&quot;&gt;
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				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;← Older revision&lt;/td&gt;
				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;Revision as of 14:15, 14 April 2008&lt;/td&gt;
				&lt;/tr&gt;&lt;tr&gt;
  &lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 1:&lt;/td&gt;
  &lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 1:&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
  &lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;
  &lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;The rope length of the trefoil is known to be no more than 16.372, by [[A ropelength bibliography|numerical experiments]], while the [[Article:Quadrisecants Give New Lower Bounds for the Ropelength of a Knot|sharpest known lower bound]] (actually applicable to all non-trivial knots) is 15.66.&lt;/div&gt;&lt;/td&gt;
  &lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;
  &lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;The rope length of the &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[3_1|&lt;/ins&gt;trefoil&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]]&lt;/ins&gt; is known to be no more than 16.372, by [[A ropelength bibliography|numerical experiments]], while the [[Article:Quadrisecants Give New Lower Bounds for the Ropelength of a Knot|sharpest known lower bound]] (actually applicable to all non-trivial knots) is 15.66.&lt;/div&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
  &lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;
  &lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br /&gt;&lt;/td&gt;
  &lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;
  &lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br /&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
  &lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;
  &lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;[[Image:The Trefoil is Fibered.png|right|120px]] The trefoil is a fibered knot! A java applet demonstrating it, written by Robert Barrington Leigh at the University of Toronto, is [http://www.math.toronto.edu/~drorbn/People/BarringtonLeigh/FiberedKnot.html here].&lt;/div&gt;&lt;/td&gt;
  &lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;
  &lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;[[Image:The Trefoil is Fibered.png|right|120px]] The trefoil is a fibered knot! A java applet demonstrating it, written by Robert Barrington Leigh at the University of Toronto, is [http://www.math.toronto.edu/~drorbn/People/BarringtonLeigh/FiberedKnot.html here].&lt;/div&gt;&lt;/td&gt;
&lt;/tr&gt;

&lt;!-- diff cache key knotsdb-mw_:diff:wikidiff2:1.12:old-120614:rev-1691719:1.13.0 --&gt;
&lt;/table&gt;</summary>
		<author><name>Drorbn</name></author>
	</entry>
	<entry>
		<id>https://katlas.org/index.php?title=Notes_for_3_1%27s_three_dimensional_invariants&amp;diff=120614&amp;oldid=prev</id>
		<title>Drorbn at 18:42, 27 July 2006</title>
		<link rel="alternate" type="text/html" href="https://katlas.org/index.php?title=Notes_for_3_1%27s_three_dimensional_invariants&amp;diff=120614&amp;oldid=prev"/>
		<updated>2006-07-27T18:42:21Z</updated>

		<summary type="html">&lt;p&gt;&lt;/p&gt;
&lt;table style=&quot;background-color: #fff; color: #202122;&quot; data-mw=&quot;interface&quot;&gt;
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				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;← Older revision&lt;/td&gt;
				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;Revision as of 14:42, 27 July 2006&lt;/td&gt;
				&lt;/tr&gt;&lt;tr&gt;
  &lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 1:&lt;/td&gt;
  &lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 1:&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
  &lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;
  &lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;The rope length of the trefoil is known to be no more than 16.372, by [[A ropelength bibliography|numerical experiments]], while the [[Article:Quadrisecants Give New Lower Bounds for the Ropelength of a Knot|sharpest known lower bound]] (actually applicable to all non-trivial knots) is 15.66.&lt;/div&gt;&lt;/td&gt;
  &lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;
  &lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;The rope length of the trefoil is known to be no more than 16.372, by [[A ropelength bibliography|numerical experiments]], while the [[Article:Quadrisecants Give New Lower Bounds for the Ropelength of a Knot|sharpest known lower bound]] (actually applicable to all non-trivial knots) is 15.66.&lt;/div&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
  &lt;td colspan=&quot;2&quot; class=&quot;diff-empty diff-side-deleted&quot;&gt;&lt;/td&gt;
  &lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;
  &lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br /&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
  &lt;td colspan=&quot;2&quot; class=&quot;diff-empty diff-side-deleted&quot;&gt;&lt;/td&gt;
  &lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;
  &lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;[[Image:The Trefoil is Fibered.png|right|120px]] The trefoil is a fibered knot! A java applet demonstrating it, written by Robert Barrington Leigh at the University of Toronto, is [http://www.math.toronto.edu/~drorbn/People/BarringtonLeigh/FiberedKnot.html here].&lt;/div&gt;&lt;/td&gt;
&lt;/tr&gt;

&lt;!-- diff cache key knotsdb-mw_:diff:wikidiff2:1.12:old-57514:rev-120614:1.13.0 --&gt;
&lt;/table&gt;</summary>
		<author><name>Drorbn</name></author>
	</entry>
	<entry>
		<id>https://katlas.org/index.php?title=Notes_for_3_1%27s_three_dimensional_invariants&amp;diff=57514&amp;oldid=prev</id>
		<title>Scott at 17:50, 23 September 2005</title>
		<link rel="alternate" type="text/html" href="https://katlas.org/index.php?title=Notes_for_3_1%27s_three_dimensional_invariants&amp;diff=57514&amp;oldid=prev"/>
		<updated>2005-09-23T17:50:31Z</updated>

		<summary type="html">&lt;p&gt;&lt;/p&gt;
&lt;table style=&quot;background-color: #fff; color: #202122;&quot; data-mw=&quot;interface&quot;&gt;
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				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;← Older revision&lt;/td&gt;
				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;Revision as of 13:50, 23 September 2005&lt;/td&gt;
				&lt;/tr&gt;&lt;tr&gt;
  &lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 1:&lt;/td&gt;
  &lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 1:&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
  &lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;
  &lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;The rope length of the trefoil is known to be no more than 16.372, by numerical experiments, while the [[Article:Quadrisecants Give New Lower Bounds for the Ropelength of a Knot|sharpest known lower bound]] (actually applicable to all non-trivial knots) is 15.66.&lt;/div&gt;&lt;/td&gt;
  &lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;
  &lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;The rope length of the trefoil is known to be no more than 16.372, by &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[A ropelength bibliography|&lt;/ins&gt;numerical experiments&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]]&lt;/ins&gt;, while the [[Article:Quadrisecants Give New Lower Bounds for the Ropelength of a Knot|sharpest known lower bound]] (actually applicable to all non-trivial knots) is 15.66.&lt;/div&gt;&lt;/td&gt;
&lt;/tr&gt;

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&lt;/table&gt;</summary>
		<author><name>Scott</name></author>
	</entry>
	<entry>
		<id>https://katlas.org/index.php?title=Notes_for_3_1%27s_three_dimensional_invariants&amp;diff=48491&amp;oldid=prev</id>
		<title>Scott at 17:18, 23 September 2005</title>
		<link rel="alternate" type="text/html" href="https://katlas.org/index.php?title=Notes_for_3_1%27s_three_dimensional_invariants&amp;diff=48491&amp;oldid=prev"/>
		<updated>2005-09-23T17:18:03Z</updated>

		<summary type="html">&lt;p&gt;&lt;/p&gt;
&lt;table style=&quot;background-color: #fff; color: #202122;&quot; data-mw=&quot;interface&quot;&gt;
				&lt;col class=&quot;diff-marker&quot; /&gt;
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				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;← Older revision&lt;/td&gt;
				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;Revision as of 13:18, 23 September 2005&lt;/td&gt;
				&lt;/tr&gt;&lt;tr&gt;
  &lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 1:&lt;/td&gt;
  &lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 1:&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
  &lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;
  &lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;The rope length of the trefoil is known to be no more than 16.372, by numerical experiments, while the [[Article:Quadrisecants Give New Lower Bounds&lt;/div&gt;&lt;/td&gt;
  &lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;
  &lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;The rope length of the trefoil is known to be no more than 16.372, by numerical experiments, while the [[Article:Quadrisecants Give New Lower Bounds&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; for the Ropelength of a Knot|sharpest known lower bound]] (actually applicable to all non-trivial knots) is 15.66.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
  &lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;
  &lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;for the Ropelength of a Knot|sharpest known lower bound]] (actually applicable to all non-trivial knots) is 15.66.&lt;/div&gt;&lt;/td&gt;
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&lt;/table&gt;</summary>
		<author><name>Scott</name></author>
	</entry>
	<entry>
		<id>https://katlas.org/index.php?title=Notes_for_3_1%27s_three_dimensional_invariants&amp;diff=48471&amp;oldid=prev</id>
		<title>Scott at 17:17, 23 September 2005</title>
		<link rel="alternate" type="text/html" href="https://katlas.org/index.php?title=Notes_for_3_1%27s_three_dimensional_invariants&amp;diff=48471&amp;oldid=prev"/>
		<updated>2005-09-23T17:17:34Z</updated>

		<summary type="html">&lt;p&gt;&lt;/p&gt;
&lt;p&gt;&lt;b&gt;New page&lt;/b&gt;&lt;/p&gt;&lt;div&gt;The rope length of the trefoil is known to be no more than 16.372, by numerical experiments, while the [[Article:Quadrisecants Give New Lower Bounds&lt;br /&gt;
for the Ropelength of a Knot|sharpest known lower bound]] (actually applicable to all non-trivial knots) is 15.66.&lt;/div&gt;</summary>
		<author><name>Scott</name></author>
	</entry>
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