https://katlas.org/index.php?action=history&feed=atom&title=Three_Dimensional_Invariants 2026-05-08T19:04:15Z Revision history for this page on the wiki MediaWiki 1.39.6 https://katlas.org/index.php?title=Three_Dimensional_Invariants&diff=1691357&oldid=prev 2007-12-19T00:49:19Z

<p><span dir="auto"><span class="autocomment">3-Genus</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:49, 18 December 2007</td> </tr><tr> <td colspan="2" class="diff-lineno">Line 182:</td> <td colspan="2" class="diff-lineno">Line 182:</td> </tr> <tr> <td class="diff-marker"></td> <td style="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;"><div>&lt;!--END--&gt;</div></td> <td class="diff-marker"></td> <td style="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;"><div>&lt;!--END--&gt;</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="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;"><br /></td> <td class="diff-marker"></td> <td style="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;"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="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;"><div>{{Knot Image Pair|K11n34|gif|<del style="font-weight: bold; text-decoration: none;">K11n32</del>|gif}}</div></td> <td class="diff-marker" data-marker="+"></td> <td style="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;"><div>{{Knot Image Pair|K11n34|gif|<ins style="font-weight: bold; text-decoration: none;">K11n42</ins>|gif}}</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="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;"><br /></td> <td class="diff-marker"></td> <td style="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;"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td style="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;"><div>====Bridge Index====</div></td> <td class="diff-marker"></td> <td style="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;"><div>====Bridge Index====</div></td> </tr> </table>

Drorbn https://katlas.org/index.php?title=Three_Dimensional_Invariants&diff=1689294&oldid=prev 2007-08-28T23:31:05Z

<p>Reverted edits by <a href="/wiki/Special:Contributions/67.155.176.189" title="Special:Contributions/67.155.176.189">67.155.176.189</a> (<a href="/index.php?title=User_talk:67.155.176.189&amp;action=edit&amp;redlink=1" class="new" title="User talk:67.155.176.189 (page does not exist)">Talk</a>); changed back to last version by <a href="/wiki/User:Scott" title="User:Scott">Scott</a></p> <a href="https://katlas.org/index.php?title=Three_Dimensional_Invariants&amp;diff=1689294&amp;oldid=1689230">Show changes</a>

Scott https://katlas.org/index.php?title=Three_Dimensional_Invariants&diff=1689230&oldid=prev 2007-08-28T16:31:23Z

<p></p> <a href="https://katlas.org/index.php?title=Three_Dimensional_Invariants&amp;diff=1689230&amp;oldid=62520">Show changes</a>

67.155.176.189 https://katlas.org/index.php?title=Three_Dimensional_Invariants&diff=62520&oldid=prev 2007-06-15T22:46:00Z

<p>Reverted edit of 62.231.243.138, changed back to last version by Scott</p> <a href="https://katlas.org/index.php?title=Three_Dimensional_Invariants&amp;diff=62520&amp;oldid=58343">Show changes</a>

Scott https://katlas.org/index.php?title=Three_Dimensional_Invariants&diff=58343&oldid=prev 2007-06-15T22:10:52Z

<p></p> <a href="https://katlas.org/index.php?title=Three_Dimensional_Invariants&amp;diff=58343&amp;oldid=58342">Show changes</a>

62.231.243.138 https://katlas.org/index.php?title=Three_Dimensional_Invariants&diff=58342&oldid=prev 2007-06-15T17:15:34Z

<p>Reverted edit of 62.231.243.138, changed back to last version by Drorbn</p> <a href="https://katlas.org/index.php?title=Three_Dimensional_Invariants&amp;diff=58342&amp;oldid=58331">Show changes</a>

Scott https://katlas.org/index.php?title=Three_Dimensional_Invariants&diff=58331&oldid=prev 2007-06-15T13:13:24Z

<p></p> <a href="https://katlas.org/index.php?title=Three_Dimensional_Invariants&amp;diff=58331&amp;oldid=58321">Show changes</a>

62.231.243.138 https://katlas.org/index.php?title=Three_Dimensional_Invariants&diff=58321&oldid=prev 2006-04-16T11:00:08Z

<p><span dir="auto"><span class="autocomment">3-Genus</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 07:00, 16 April 2006</td> </tr><tr> <td colspan="2" class="diff-lineno">Line 127:</td> <td colspan="2" class="diff-lineno">Line 127:</td> </tr> <tr> <td class="diff-marker"></td> <td style="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;"><br /></td> <td class="diff-marker"></td> <td style="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;"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td style="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;"><div>A Seifert surface for a knot &lt;math&gt;K \subset S^3&lt;/math&gt; is a compact oriented surface &lt;math&gt;L \subset S^3&lt;/math&gt;</div></td> <td class="diff-marker"></td> <td style="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;"><div>A Seifert surface for a knot &lt;math&gt;K \subset S^3&lt;/math&gt; is a compact oriented surface &lt;math&gt;L \subset S^3&lt;/math&gt;</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="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;"><div>with boundary &lt;math&gt;\partial L=K&lt;/math&gt;. Seifert surfaces exist, but are not unique. The SeifertView programme</div></td> <td class="diff-marker" data-marker="+"></td> <td style="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;"><div>with boundary &lt;math&gt;\partial L=K&lt;/math&gt;. Seifert surfaces exist, but are not unique. The<ins style="font-weight: bold; text-decoration: none;"> [http://www.win.tue.nl/~vanwijk/seifertview/</ins> SeifertView programme<ins style="font-weight: bold; text-decoration: none;">] is a visual implementation of the algorithm of Seifert (1934) for </ins></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="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;"><div>http://www.win.tue.nl/~vanwijk/seifertview/ is a visual implementation of the algorithm of Seifert (1934) for </div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td style="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;"><div>the construction of a Seifert surface from a knot projection. The 3-genus of a knot is the minimal genus of a </div></td> <td class="diff-marker"></td> <td style="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;"><div>the construction of a Seifert surface from a knot projection. The 3-genus of a knot is the minimal genus of a </div></td> </tr> <tr> <td class="diff-marker"></td> <td style="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;"><div>Seifert surface for that knot.</div></td> <td class="diff-marker"></td> <td style="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;"><div>Seifert surface for that knot.</div></td> </tr> </table>

Drorbn https://katlas.org/index.php?title=Three_Dimensional_Invariants&diff=49687&oldid=prev 2006-04-16T10:55:53Z

<p><span dir="auto"><span class="autocomment">3-Genus</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 06:55, 16 April 2006</td> </tr><tr> <td colspan="2" class="diff-lineno">Line 126:</td> <td colspan="2" class="diff-lineno">Line 126:</td> </tr> <tr> <td class="diff-marker"></td> <td style="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;"><div>====3-Genus====</div></td> <td class="diff-marker"></td> <td style="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;"><div>====3-Genus====</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="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;"><br /></td> <td class="diff-marker"></td> <td style="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;"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="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;"><div><del style="font-weight: bold; text-decoration: none;">The</del> <del style="font-weight: bold; text-decoration: none;">3-genus</del> <del style="font-weight: bold; text-decoration: none;">of</del> a knot <del style="font-weight: bold; text-decoration: none;">is</del> <del style="font-weight: bold; text-decoration: none;">the</del> <del style="font-weight: bold; text-decoration: none;">minimal</del> <del style="font-weight: bold; text-decoration: none;">genus of</del> a <del style="font-weight: bold; text-decoration: none;">Seifert</del> surface <del style="font-weight: bold; text-decoration: none;">for</del> <del style="font-weight: bold; text-decoration: none;">that</del> <del style="font-weight: bold; text-decoration: none;">knot.</del></div></td> <td class="diff-marker" data-marker="+"></td> <td style="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;"><div><ins style="font-weight: bold; text-decoration: none;">A</ins> <ins style="font-weight: bold; text-decoration: none;">Seifert</ins> <ins style="font-weight: bold; text-decoration: none;">surface for</ins> a knot <ins style="font-weight: bold; text-decoration: none;">&lt;math&gt;K</ins> <ins style="font-weight: bold; text-decoration: none;">\subset</ins> <ins style="font-weight: bold; text-decoration: none;">S^3&lt;/math&gt;</ins> <ins style="font-weight: bold; text-decoration: none;">is</ins> a <ins style="font-weight: bold; text-decoration: none;">compact oriented</ins> surface <ins style="font-weight: bold; text-decoration: none;">&lt;math&gt;L</ins> <ins style="font-weight: bold; text-decoration: none;">\subset</ins> <ins style="font-weight: bold; text-decoration: none;">S^3&lt;/math&gt;</ins></div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td style="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;"><div>with boundary &lt;math&gt;\partial L=K&lt;/math&gt;. Seifert surfaces exist, but are not unique. The SeifertView programme</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td style="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;"><div>http://www.win.tue.nl/~vanwijk/seifertview/ is a visual implementation of the algorithm of Seifert (1934) for </div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td style="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;"><div>the construction of a Seifert surface from a knot projection. The 3-genus of a knot is the minimal genus of a </div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td style="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;"><div>Seifert surface for that knot.</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td style="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;"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td style="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;"><br /></td> <td class="diff-marker"></td> <td style="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;"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td style="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;"><div>&lt;!--$$?ThreeGenus$$--&gt;</div></td> <td class="diff-marker"></td> <td style="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;"><div>&lt;!--$$?ThreeGenus$$--&gt;</div></td> </tr> </table>

Ranicki https://katlas.org/index.php?title=Three_Dimensional_Invariants&diff=49685&oldid=prev 2006-02-07T19:23:52Z

<p><span dir="auto"><span class="autocomment">3-Genus</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:23, 7 February 2006</td> </tr><tr> <td colspan="2" class="diff-lineno">Line 178:</td> <td colspan="2" class="diff-lineno">Line 178:</td> </tr> <tr> <td class="diff-marker"></td> <td style="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;"><div>&lt;!--END--&gt;</div></td> <td class="diff-marker"></td> <td style="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;"><div>&lt;!--END--&gt;</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="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;"><br /></td> <td class="diff-marker"></td> <td style="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;"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td style="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;"><div>{{Knot Image Pair|<del style="font-weight: bold; text-decoration: none;">K11a34</del>|gif|K11n32|gif}}</div></td> <td class="diff-marker" data-marker="+"></td> <td style="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;"><div>{{Knot Image Pair|<ins style="font-weight: bold; text-decoration: none;">K11n34</ins>|gif|K11n32|gif}}</div></td> </tr> <tr> <td class="diff-marker"></td> <td style="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;"><br /></td> <td class="diff-marker"></td> <td style="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;"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td style="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;"><div>====Bridge Index====</div></td> <td class="diff-marker"></td> <td style="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;"><div>====Bridge Index====</div></td> </tr> </table>

Drorbn