https://drorbn.net/index.php?action=history&feed=atom&title=WKO
WKO - Revision history
2026-06-20T18:02:01Z
Revision history for this page on the wiki
MediaWiki 1.39.6
https://drorbn.net/index.php?title=WKO&diff=13184&oldid=prev
Drorbn at 21:47, 5 May 2014
2014-05-05T21:47:37Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:47, 5 May 2014</td>
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<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><span style="color:red">This paper was split in two and became the first two parts of a four-part series ({{Home link|LOP.html#WKO1|WKO1}}, {{Home link|LOP.html#WKO2|WKO2}}, {{Pensieve link|Projects/WKO3/|WKO3}}, {{Pensieve link|Projects/WKO4/|WKO4}}). The remaining relevance of this page is due to the series of videotaped lectures (wClips) that are linked here.</span></div></td>
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<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><span style="color:red">This paper was split in two and became the first two parts of a four-part series ({{Home link|LOP.html#WKO1|WKO1}}, {{Home link|LOP.html#WKO2|WKO2}}, {{Pensieve link|Projects/WKO3/|WKO3}}, {{Pensieve link|Projects/WKO4/|WKO4}}). The remaining relevance of this page is due to the series of videotaped lectures (wClips) that are linked here.</span></div></td>
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<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>
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<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>
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<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><span style="color:red"><b>Download</b></span> [http://www.math.toronto.edu/~drorbn/papers/WKO/WKO.pdf WKO.pdf]: last updated &ge; <del style="font-weight: bold; text-decoration: none;">November</del> <del style="font-weight: bold; text-decoration: none;">13</del>, <del style="font-weight: bold; text-decoration: none;">2013</del>. first edition: September 27, 2013. The {{arXiv|1309.7155}} edition may be older.</div></td>
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<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><span style="color:red"><b>Download</b></span> [http://www.math.toronto.edu/~drorbn/papers/WKO/WKO.pdf WKO.pdf]: last updated &ge; <ins style="font-weight: bold; text-decoration: none;">May</ins> <ins style="font-weight: bold; text-decoration: none;">5</ins>, <ins style="font-weight: bold; text-decoration: none;">2014</ins>. first edition: September 27, 2013. The {{arXiv|1309.7155}} edition may be older.</div></td>
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<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>
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<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>
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<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>'''Abstract.''' w-Knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.) make a class of knotted objects which is <u>w</u>ider but <u>w</u>eaker than their "<u>u</u>sual" counterparts. To get (say) w-knots from u-knots, one has to allow non-planar "virtual" knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation, the "overcrossings commute" relation, further beyond the ordinary collection of Reidemeister moves, making w-knotted objects a bit weaker once again.</div></td>
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<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>'''Abstract.''' w-Knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.) make a class of knotted objects which is <u>w</u>ider but <u>w</u>eaker than their "<u>u</u>sual" counterparts. To get (say) w-knots from u-knots, one has to allow non-planar "virtual" knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation, the "overcrossings commute" relation, further beyond the ordinary collection of Reidemeister moves, making w-knotted objects a bit weaker once again.</div></td>
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Drorbn
https://drorbn.net/index.php?title=WKO&diff=13183&oldid=prev
Drorbn at 21:46, 5 May 2014
2014-05-05T21:46:40Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:46, 5 May 2014</td>
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<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>Joint with [http://www.math.toronto.edu/zsuzsi/ Zsuzsanna Dancso]</div></td>
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<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>Joint with [http://www.math.toronto.edu/zsuzsi/ Zsuzsanna Dancso]</div></td>
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<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>
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<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><span style="color:red">This paper was split in two and became the first two parts of a four-part series ({{Home link|LOP.html#WKO1|WKO1}}, {{Home link|LOP.html#WKO2|WKO2}}, {{Pensieve link|Projects/WKO3/|WKO3}}, {{Pensieve link|Projects/WKO4/|WKO4}}). The remaining relevance of this page is due to the series of videotaped lectures (wClips) that are linked here.</span></div></td>
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<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>
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<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>
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<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><span style="color:red"><b>Download</b></span> [http://www.math.toronto.edu/~drorbn/papers/WKO/WKO.pdf WKO.pdf]: last updated &ge; November 13, 2013. first edition: September 27, 2013. The {{arXiv|1309.7155}} edition may be older.</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><span style="color:red"><b>Download</b></span> [http://www.math.toronto.edu/~drorbn/papers/WKO/WKO.pdf WKO.pdf]: last updated &ge; November 13, 2013. first edition: September 27, 2013. The {{arXiv|1309.7155}} edition may be older.</div></td>
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Drorbn
https://drorbn.net/index.php?title=WKO&diff=12988&oldid=prev
Drorbn at 20:11, 13 November 2013
2013-11-13T20:11:03Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:11, 13 November 2013</td>
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<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>Joint with [http://www.math.toronto.edu/zsuzsi/ Zsuzsanna Dancso]</div></td>
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<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>
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<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><span style="color:red"><b>Download</b></span> [http://www.math.toronto.edu/~drorbn/papers/WKO/WKO.pdf WKO.pdf]: last updated &ge; <del style="font-weight: bold; text-decoration: none;">September</del> <del style="font-weight: bold; text-decoration: none;">30</del>, 2013. first edition: September 27, 2013. The {{arXiv|1309.7155}} edition may be older.</div></td>
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<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><span style="color:red"><b>Download</b></span> [http://www.math.toronto.edu/~drorbn/papers/WKO/WKO.pdf WKO.pdf]: last updated &ge; <ins style="font-weight: bold; text-decoration: none;">November</ins> <ins style="font-weight: bold; text-decoration: none;">13</ins>, 2013. first edition: September 27, 2013. The {{arXiv|1309.7155}} edition may be older.</div></td>
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<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>
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<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>'''Abstract.''' w-Knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.) make a class of knotted objects which is <u>w</u>ider but <u>w</u>eaker than their "<u>u</u>sual" counterparts. To get (say) w-knots from u-knots, one has to allow non-planar "virtual" knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation, the "overcrossings commute" relation, further beyond the ordinary collection of Reidemeister moves, making w-knotted objects a bit weaker once again.</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>'''Abstract.''' w-Knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.) make a class of knotted objects which is <u>w</u>ider but <u>w</u>eaker than their "<u>u</u>sual" counterparts. To get (say) w-knots from u-knots, one has to allow non-planar "virtual" knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation, the "overcrossings commute" relation, further beyond the ordinary collection of Reidemeister moves, making w-knotted objects a bit weaker once again.</div></td>
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Drorbn
https://drorbn.net/index.php?title=WKO&diff=12986&oldid=prev
Drorbn at 17:16, 13 November 2013
2013-11-13T17:16:41Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:16, 13 November 2013</td>
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<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 true value of w-knots, though, is likely to emerge later, for we expect them to serve as a <u>w</u>armup example for what we expect will be even more interesting - the study of <u>v</u>irtual knots, or v-knots. We expect v-knotted objects to provide the global context whose projectivization (or "associated graded structure") will be the Etingof-Kazhdan theory of deformation quantization of Lie bialgebras {{ref|EK}}. </div></td>
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<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 true value of w-knots, though, is likely to emerge later, for we expect them to serve as a <u>w</u>armup example for what we expect will be even more interesting - the study of <u>v</u>irtual knots, or v-knots. We expect v-knotted objects to provide the global context whose projectivization (or "associated graded structure") will be the Etingof-Kazhdan theory of deformation quantization of Lie bialgebras {{ref|EK}}. </div></td>
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<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>'''The paper.'''<del style="font-weight: bold; text-decoration: none;"> . [http://www.math.toronto.edu/zsuzsi/research/WKO/WKO.pdf WKO.pdf], [http://www.math.toronto.edu/zsuzsi/research/WKO/WKO.zip WKO.zip] (Dror's version:</del> {{Home Link|papers/WKO/WKO.pdf|WKO.pdf}}, {{Home Link|papers/WKO/WKO.zip|WKO.zip}}<del style="font-weight: bold; text-decoration: none;">)</del>.</div></td>
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<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 paper.''' {{Home Link|papers/WKO/WKO.pdf|WKO.pdf}}, {{Home Link|papers/WKO/WKO.zip|WKO.zip}}.</div></td>
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<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>
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<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>'''Related Mathematica Notebooks.''' "The Kishino Braid" ({{Pensieve Link|Projects/WKO/The_Kishino_Braid.nb|Source}}, {{Pensieve Link|Projects/WKO/nb/The_Kishino_Braid.pdf|PDF}}), "Dimensions" ({{Pensieve Link|Projects/WKO/Dimensions.nb|Source}}, {{Pensieve Link|Projects/WKO/nb/Dimensions|PDF}}), "wA" ({{Pensieve Link|Projects/WKO/wA.nb|Source}}, {{Pensieve Link|Projects/WKO/nb/wA.pdf|PDF}}), "InfinitesimalAlexanderModules" ({{Pensieve Link|Projects/WKO/InfinitesimalAlexanderModules.nb|Source}}, {{Pensieve Link|Projects/WKO/nb/InfinitesimalAlexanderModules.pdf|PDF}}).</div></td>
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<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>'''Related Mathematica Notebooks.''' "The Kishino Braid" ({{Pensieve Link|Projects/WKO/The_Kishino_Braid.nb|Source}}, {{Pensieve Link|Projects/WKO/nb/The_Kishino_Braid.pdf|PDF}}), "Dimensions" ({{Pensieve Link|Projects/WKO/Dimensions.nb|Source}}, {{Pensieve Link|Projects/WKO/nb/Dimensions|PDF}}), "wA" ({{Pensieve Link|Projects/WKO/wA.nb|Source}}, {{Pensieve Link|Projects/WKO/nb/wA.pdf|PDF}}), "InfinitesimalAlexanderModules" ({{Pensieve Link|Projects/WKO/InfinitesimalAlexanderModules.nb|Source}}, {{Pensieve Link|Projects/WKO/nb/InfinitesimalAlexanderModules.pdf|PDF}}).</div></td>
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Drorbn
https://drorbn.net/index.php?title=WKO&diff=12978&oldid=prev
Drorbn at 07:32, 30 September 2013
2013-09-30T07:32:54Z
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<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>{{wClips/Navigation}}</div></td>
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<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>
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<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>
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<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>==Finite Type Invariants of W-Knotted Objects: From Alexander to Kashiwara and Vergne==</div></td>
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<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>==Finite Type Invariants of W-Knotted Objects: From Alexander to Kashiwara and Vergne==</div></td>
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<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>Joint with [http://www.math.toronto.edu/zsuzsi/ Zsuzsanna Dancso]</div></td>
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<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>
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<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>
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<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><span style="color:red"><b>Download</b></span> [http://www.math.toronto.edu/~drorbn/papers/WKO/WKO.pdf WKO.pdf]: last updated &ge; September <del style="font-weight: bold; text-decoration: none;">27</del>, 2013. first edition: September 27, 2013.</div></td>
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<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><span style="color:red"><b>Download</b></span> [http://www.math.toronto.edu/~drorbn/papers/WKO/WKO.pdf WKO.pdf]: last updated &ge; September <ins style="font-weight: bold; text-decoration: none;">30</ins>, 2013. first edition: September 27, 2013<ins style="font-weight: bold; text-decoration: none;">. The {{arXiv|1309.7155}} edition may be older</ins>.</div></td>
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<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>
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<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>
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<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>'''Abstract.''' w-Knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.) make a class of knotted objects which is <u>w</u>ider but <u>w</u>eaker than their "<u>u</u>sual" counterparts. To get (say) w-knots from u-knots, one has to allow non-planar "virtual" knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation, the "overcrossings commute" relation, further beyond the ordinary collection of Reidemeister moves, making w-knotted objects a bit weaker once again.</div></td>
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<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>'''Abstract.''' w-Knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.) make a class of knotted objects which is <u>w</u>ider but <u>w</u>eaker than their "<u>u</u>sual" counterparts. To get (say) w-knots from u-knots, one has to allow non-planar "virtual" knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation, the "overcrossings commute" relation, further beyond the ordinary collection of Reidemeister moves, making w-knotted objects a bit weaker once again.</div></td>
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Drorbn
https://drorbn.net/index.php?title=WKO&diff=12977&oldid=prev
Drorbn at 09:53, 27 September 2013
2013-09-27T09:53:05Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 05:53, 27 September 2013</td>
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<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>Joint with [http://www.math.toronto.edu/zsuzsi/ Zsuzsanna Dancso]</div></td>
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<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>
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<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><span style="color:red"><b>Download</b></span> [http://www.math.toronto.edu/~drorbn/papers/WKO/WKO.pdf WKO.pdf]: last updated &ge; <del style="font-weight: bold; text-decoration: none;">August</del> <del style="font-weight: bold; text-decoration: none;">8</del>, 2013. first edition: <del style="font-weight: bold; text-decoration: none;">not</del> <del style="font-weight: bold; text-decoration: none;">yet</del>.</div></td>
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<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><span style="color:red"><b>Download</b></span> [http://www.math.toronto.edu/~drorbn/papers/WKO/WKO.pdf WKO.pdf]: last updated &ge; <ins style="font-weight: bold; text-decoration: none;">September</ins> <ins style="font-weight: bold; text-decoration: none;">27</ins>, 2013. first edition: <ins style="font-weight: bold; text-decoration: none;">September</ins> <ins style="font-weight: bold; text-decoration: none;">27, 2013</ins>.</div></td>
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<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>
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<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>
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<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>'''Abstract.''' w-Knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.) make a class of knotted objects which is <u>w</u>ider but <u>w</u>eaker than their "<u>u</u>sual" counterparts. To get (say) w-knots from u-knots, one has to allow non-planar "virtual" knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation, the "overcrossings commute" relation, further beyond the ordinary collection of Reidemeister moves, making w-knotted objects a bit weaker once again.</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>'''Abstract.''' w-Knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.) make a class of knotted objects which is <u>w</u>ider but <u>w</u>eaker than their "<u>u</u>sual" counterparts. To get (say) w-knots from u-knots, one has to allow non-planar "virtual" knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation, the "overcrossings commute" relation, further beyond the ordinary collection of Reidemeister moves, making w-knotted objects a bit weaker once again.</div></td>
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Drorbn
https://drorbn.net/index.php?title=WKO&diff=12976&oldid=prev
Drorbn at 08:41, 27 September 2013
2013-09-27T08:41:40Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 04:41, 27 September 2013</td>
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<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>In this article we study finite type invariants of several classes of w-knotted objects. Following Berceanu and Papadima {{ref|BP}}, we construct homomorphic universal finite type invariants of w-braids and of w-tangles. We find that the universal finite type invariant of w-knots is more or less the Alexander polynomial (details inside).</div></td>
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<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>In this article we study finite type invariants of several classes of w-knotted objects. Following Berceanu and Papadima {{ref|BP}}, we construct homomorphic universal finite type invariants of w-braids and of w-tangles. We find that the universal finite type invariant of w-knots is more or less the Alexander polynomial (details inside).</div></td>
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<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>
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<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>
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<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>Much as the spaces <math>{\mathcal A}</math> of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, we find that the spaces <math>{\mathcal A}^w</math> of "arrow diagrams" for w-knotted objects are related to not-necessarily-metrized Lie algebras. Many questions concerning w-knotted objects turn out to be equivalent to questions about Lie algebras. Most notably we find that a homomorphic universal finite type invariant of w-knotted <del style="font-weight: bold; text-decoration: none;">trivalent graphs</del> is essentially the same as a solution of the Kashiwara-Vergne {{ref|KV}} conjecture and much of the Alekseev-Torossian {{ref|AT}} work on Drinfel'd associators and Kashiwara-Vergne can be re-interpreted as a study of w-knotted trivalent graphs.</div></td>
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<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>Much as the spaces <math>{\mathcal A}</math> of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, we find that the spaces <math>{\mathcal A}^w</math> of "arrow diagrams" for w-knotted objects are related to not-necessarily-metrized Lie algebras. Many questions concerning w-knotted objects turn out to be equivalent to questions about Lie algebras. Most notably we find that a homomorphic universal finite type invariant of w-knotted <ins style="font-weight: bold; text-decoration: none;">foams</ins> is essentially the same as a solution of the Kashiwara-Vergne {{ref|KV}} conjecture and much of the Alekseev-Torossian {{ref|AT}} work on Drinfel'd associators and Kashiwara-Vergne can be re-interpreted as a study of w-knotted trivalent graphs.</div></td>
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<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>
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<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>
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<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 true value of w-knots, though, is likely to emerge later, for we expect them to serve as a <u>w</u>armup example for what we expect will be even more interesting - the study of <u>v</u>irtual knots, or v-knots. We expect v-knotted objects to provide the global context whose projectivization (or "associated graded structure") will be the Etingof-Kazhdan theory of deformation quantization of Lie bialgebras {{ref|EK}}. </div></td>
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<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 true value of w-knots, though, is likely to emerge later, for we expect them to serve as a <u>w</u>armup example for what we expect will be even more interesting - the study of <u>v</u>irtual knots, or v-knots. We expect v-knotted objects to provide the global context whose projectivization (or "associated graded structure") will be the Etingof-Kazhdan theory of deformation quantization of Lie bialgebras {{ref|EK}}. </div></td>
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Drorbn
https://drorbn.net/index.php?title=WKO&diff=12975&oldid=prev
Drorbn at 11:59, 26 September 2013
2013-09-26T11:59:24Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 07:59, 26 September 2013</td>
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<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 group of w-braids was studied (under the name "<u>w</u>elded braids") by Fenn, Rimanyi and Rourke {{ref|FRR}} and was shown to be isomorphic to the McCool group {{ref|Mc}} of "basis-conjugating" automorphisms of a free group <math>F_n</math> - the smallest subgroup of <math>\operatorname{Aut}(F_n)</math> that contains both braids and permutations. Brendle and Hatcher {{ref|BH}}, in work that traces back to Goldsmith {{ref|Gol}}, have shown this group to be a group of movies of flying rings in <math>{\mathbb R}^3</math>. Satoh {{ref|Sa}} studied several classes of w-knotted objects (under the name "<u>w</u>eakly-virtual") and has shown them to be closely related to certain classes of knotted surfaces in <math>{\mathbb R}^4</math>. So w-knotted objects are algebraically and topologically interesting.</div></td>
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<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 group of w-braids was studied (under the name "<u>w</u>elded braids") by Fenn, Rimanyi and Rourke {{ref|FRR}} and was shown to be isomorphic to the McCool group {{ref|Mc}} of "basis-conjugating" automorphisms of a free group <math>F_n</math> - the smallest subgroup of <math>\operatorname{Aut}(F_n)</math> that contains both braids and permutations. Brendle and Hatcher {{ref|BH}}, in work that traces back to Goldsmith {{ref|Gol}}, have shown this group to be a group of movies of flying rings in <math>{\mathbb R}^3</math>. Satoh {{ref|Sa}} studied several classes of w-knotted objects (under the name "<u>w</u>eakly-virtual") and has shown them to be closely related to certain classes of knotted surfaces in <math>{\mathbb R}^4</math>. So w-knotted objects are algebraically and topologically interesting.</div></td>
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<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>
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<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>
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<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>In this article we study finite type invariants of several classes of w-knotted objects. Following Berceanu and Papadima {{ref|BP}}, we construct<del style="font-weight: bold; text-decoration: none;"> a</del> homomorphic universal finite type <del style="font-weight: bold; text-decoration: none;">invariant</del> of w-braids<del style="font-weight: bold; text-decoration: none;">,</del> and<del style="font-weight: bold; text-decoration: none;"> hence show that the McCool group of automorphisms is "1-formal". We also construct a homomorphic universal finite type invariant</del> of w-tangles. We find that the universal finite type invariant of w-knots is more or less the Alexander polynomial (details inside).</div></td>
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<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>In this article we study finite type invariants of several classes of w-knotted objects. Following Berceanu and Papadima {{ref|BP}}, we construct homomorphic universal finite type <ins style="font-weight: bold; text-decoration: none;">invariants</ins> of w-braids and of w-tangles. We find that the universal finite type invariant of w-knots is more or less the Alexander polynomial (details inside).</div></td>
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<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>
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<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>
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<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>Much as the spaces <math>{\mathcal A}</math> of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, we find that the spaces <math>{\mathcal A}^w</math> of "arrow diagrams" for w-knotted objects are related to not-necessarily-metrized Lie algebras. Many questions concerning w-knotted objects turn out to be equivalent to questions about Lie algebras. Most notably we find that a homomorphic universal finite type invariant of w-knotted trivalent graphs is essentially the same as a solution of the Kashiwara-Vergne {{ref|KV}} conjecture and much of the Alekseev-Torossian {{ref|AT}} work on Drinfel'd associators and Kashiwara-Vergne can be re-interpreted as a study of w-knotted trivalent graphs.</div></td>
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<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>Much as the spaces <math>{\mathcal A}</math> of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, we find that the spaces <math>{\mathcal A}^w</math> of "arrow diagrams" for w-knotted objects are related to not-necessarily-metrized Lie algebras. Many questions concerning w-knotted objects turn out to be equivalent to questions about Lie algebras. Most notably we find that a homomorphic universal finite type invariant of w-knotted trivalent graphs is essentially the same as a solution of the Kashiwara-Vergne {{ref|KV}} conjecture and much of the Alekseev-Torossian {{ref|AT}} work on Drinfel'd associators and Kashiwara-Vergne can be re-interpreted as a study of w-knotted trivalent graphs.</div></td>
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Drorbn
https://drorbn.net/index.php?title=WKO&diff=12940&oldid=prev
Drorbn at 16:07, 8 August 2013
2013-08-08T16:07:57Z
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<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>Joint with [http://www.math.toronto.edu/zsuzsi/ Zsuzsanna Dancso]</div></td>
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<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>
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<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><span style="color:red"><b>Download</b></span> [http://www.math.toronto.edu/<del style="font-weight: bold; text-decoration: none;">zsuzsi</del>/<del style="font-weight: bold; text-decoration: none;">research</del>/WKO/WKO.pdf WKO.pdf]: last updated &ge; <del style="font-weight: bold; text-decoration: none;">July</del> <del style="font-weight: bold; text-decoration: none;">16</del>, 2013. first edition: not yet.</div></td>
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<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><span style="color:red"><b>Download</b></span> [http://www.math.toronto.edu/<ins style="font-weight: bold; text-decoration: none;">~drorbn</ins>/<ins style="font-weight: bold; text-decoration: none;">papers</ins>/WKO/WKO.pdf WKO.pdf]: last updated &ge; <ins style="font-weight: bold; text-decoration: none;">August</ins> <ins style="font-weight: bold; text-decoration: none;">8</ins>, 2013. first edition: not yet.</div></td>
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<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>
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<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>
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<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>'''Abstract.''' w-Knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.) make a class of knotted objects which is <u>w</u>ider but <u>w</u>eaker than their "<u>u</u>sual" counterparts. To get (say) w-knots from u-knots, one has to allow non-planar "virtual" knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation, the "overcrossings commute" relation, further beyond the ordinary collection of Reidemeister moves, making w-knotted objects a bit weaker once again.</div></td>
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<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>'''Abstract.''' w-Knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.) make a class of knotted objects which is <u>w</u>ider but <u>w</u>eaker than their "<u>u</u>sual" counterparts. To get (say) w-knots from u-knots, one has to allow non-planar "virtual" knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation, the "overcrossings commute" relation, further beyond the ordinary collection of Reidemeister moves, making w-knotted objects a bit weaker once again.</div></td>
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Drorbn
https://drorbn.net/index.php?title=WKO&diff=12934&oldid=prev
Drorbn at 14:20, 17 July 2013
2013-07-17T14:20:07Z
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<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>Joint with [http://www.math.toronto.edu/zsuzsi/ Zsuzsanna Dancso]</div></td>
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<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><span style="color:red"><b>Download</b></span> [http://www.math.toronto.edu/zsuzsi/research/WKO/WKO.pdf WKO.pdf]: last updated &ge; <del style="font-weight: bold; text-decoration: none;">May</del> <del style="font-weight: bold; text-decoration: none;">10</del>, 2013. first edition: not yet.</div></td>
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<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><span style="color:red"><b>Download</b></span> [http://www.math.toronto.edu/zsuzsi/research/WKO/WKO.pdf WKO.pdf]: last updated &ge; <ins style="font-weight: bold; text-decoration: none;">July</ins> <ins style="font-weight: bold; text-decoration: none;">16</ins>, 2013. first edition: not yet.</div></td>
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<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>
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<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>
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<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>'''Abstract.''' w-Knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.) make a class of knotted objects which is <u>w</u>ider but <u>w</u>eaker than their "<u>u</u>sual" counterparts. To get (say) w-knots from u-knots, one has to allow non-planar "virtual" knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation, the "overcrossings commute" relation, further beyond the ordinary collection of Reidemeister moves, making w-knotted objects a bit weaker once again.</div></td>
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<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>'''Abstract.''' w-Knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.) make a class of knotted objects which is <u>w</u>ider but <u>w</u>eaker than their "<u>u</u>sual" counterparts. To get (say) w-knots from u-knots, one has to allow non-planar "virtual" knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation, the "overcrossings commute" relation, further beyond the ordinary collection of Reidemeister moves, making w-knotted objects a bit weaker once again.</div></td>
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Drorbn