Revision as of 02:08, 10 May 2013

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Joint with [http://www.math.toronto.edu/zsuzsi/ Zsuzsanna Dancso]

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Download {{Home Link|papers/WKO/WKO.pdf|WKO.pdf}}: last updated ≥ March 3, 2012. first edition: not yet.

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Download [http://www.math.toronto.edu/zsuzsi/research/WKO/WKO.pdf WKO.pdf]: last updated ≥ May 10, 2013. first edition: not yet.

'''Abstract.''' w-Knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.) make a class of knotted objects which is wider but weaker than their "usual" 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.

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The true value of w-knots, though, is likely to emerge later, for we expect them to serve as a warmup example for what we expect will be even more interesting - the study of virtual 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}}.

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'''The paper.''' {{Home Link|papers/WKO/WKO.pdf|WKO.pdf}}, {{Home Link|papers/WKO/WKO.zip|WKO.zip}}. (Zsuzsi's version: [http://www.math.toronto.edu/zsuzsi/research/WKO/WKO.pdf WKO.pdf], [http://www.math.toronto.edu/zsuzsi/research/WKO/WKO.zip WKO.zip]).

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'''The paper.''' . [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: {{Home Link|papers/WKO/WKO.pdf|WKO.pdf}}, {{Home Link|papers/WKO/WKO.zip|WKO.zip}}).

'''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}}).

 I will be giving an ''Algebraic Knot Theory'' class at the University of Toronto in the spring semester of 2014, and this will be its home page.
+'''Agenda.''' Understand "(u, v, and w knots) x (topology, combinatorics, low algebra, and high algebra)". Understand the promise and the difficulty of the not-yet-existent "Algebraic Knot Theory".
+'''Description.''' There are many types of "knots", and many things to do with them. Specifically, we will talk about three master-types of knots - "u" knots which are just the usual knots we are used to seeing in 3-space, "w" knots which are 2-dimensional knots in 4-space, and "v" knots which are "virtual" knots, algebraic creatures that are not specifically embedded anywhere. Each of these master-types comes in several shades - there are plain uvw-knots, and uvw-tangles, and uvw-braids, and uvw-knotted-graphs, and there is a rich world of operations that can be applied to these, and a rich world of properties and questions to ask. And then there is the projectivization machine, which converts all these to combinatorial objects, and then algebraic (of two classes, low and high). The "Fundamental Problem" is always to construct a good map (technically, a "homomorphic expansion") from topology to combinatorics/algebra, and in the cases we understand, the solution of the Fundamental Problem involves matters such as the Knizhnik–Zamolodchikov connection, topological quantum field theory and Feynman diagrams and configuration space integrals, Drinfel'd associators and the pentagons and hexagons of category theory, and deep Lie theory. Everything we will mention, but not everything we will cover as deeply as I'd like to.
+'''References.'''
+* ''Introduction to Vassiliev Knot Invariants,'' by S. Chmutov, S. Duzhin, and J. Mostovoy, Cambridge University Press, Cambridge UK, 2012.
+* My own papers and talks and classes as can be found at http://www.math.toronto.edu/~drorbn/.
+'''Prerequisites.''' A high level of comfort with vector spaces and algebras and things you can do with them - quotients, tensor products, duals, etc. Basic topology - fundamental groups and van Kampen and rarely a bit of homology. Basic differential geometry - especially differential forms and Stokes' formula. A basic knowledge of Lie groups and algebras and their representations and an appreciation of their value. You will manage with some of that missing, but the more you will be missing the more lost you will be at times.

Drorbn - Recent changes [en]

WKO

AKT-14

 Joint with [http://www.math.toronto.edu/zsuzsi/ Zsuzsanna Dancso]
-<span style="color:red"><b>Download</b></span> {{Home Link|papers/WKO/WKO.pdf|WKO.pdf}}: last updated &ge; March 3, 2012. first edition: not yet.
+<span style="color:red"><b>Download</b></span> [http://www.math.toronto.edu/zsuzsi/research/WKO/WKO.pdf WKO.pdf]: last updated &ge; May 10, 2013. first edition: not yet.
 '''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.
 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}}.
-'''The paper.''' {{Home Link|papers/WKO/WKO.pdf|WKO.pdf}}, {{Home Link|papers/WKO/WKO.zip|WKO.zip}}. (Zsuzsi's version: [http://www.math.toronto.edu/zsuzsi/research/WKO/WKO.pdf WKO.pdf], [http://www.math.toronto.edu/zsuzsi/research/WKO/WKO.zip WKO.zip]).
+'''The paper.''' . [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: {{Home Link|papers/WKO/WKO.pdf|WKO.pdf}}, {{Home Link|papers/WKO/WKO.zip|WKO.zip}}).
 '''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}}).