WKO
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Finite Type Invariants of WKnotted Objects: From Alexander to Kashiwara and Vergne
Joint with Zsuzsanna Dancso
This paper was split in two and became the first two parts of a fourpart series (WKO1, WKO2, WKO3, WKO4). The remaining relevance of this page is due to the series of videotaped lectures (wClips) that are linked here.
Download WKO.pdf: last updated ≥ November 13, 2013. first edition: September 27, 2013. The arXiv:1309.7155 edition may be older.
Abstract. wKnots, and more generally, wknotted objects (wbraids, wtangles, etc.) make a class of knotted objects which is wider but weaker than their "usual" counterparts. To get (say) wknots from uknots, one has to allow nonplanar "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 wknotted objects a bit weaker once again.
The group of wbraids was studied (under the name "welded braids") by Fenn, Rimanyi and Rourke [FRR] and was shown to be isomorphic to the McCool group [Mc] of "basisconjugating" automorphisms of a free group  the smallest subgroup of that contains both braids and permutations. Brendle and Hatcher [BH], in work that traces back to Goldsmith [Gol], have shown this group to be a group of movies of flying rings in . Satoh [Sa] studied several classes of wknotted objects (under the name "weaklyvirtual") and has shown them to be closely related to certain classes of knotted surfaces in . So wknotted objects are algebraically and topologically interesting.
In this article we study finite type invariants of several classes of wknotted objects. Following Berceanu and Papadima [BP], we construct homomorphic universal finite type invariants of wbraids and of wtangles. We find that the universal finite type invariant of wknots is more or less the Alexander polynomial (details inside).
Much as the spaces of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, we find that the spaces of "arrow diagrams" for wknotted objects are related to notnecessarilymetrized Lie algebras. Many questions concerning wknotted objects turn out to be equivalent to questions about Lie algebras. Most notably we find that a homomorphic universal finite type invariant of wknotted foams is essentially the same as a solution of the KashiwaraVergne [KV] conjecture and much of the AlekseevTorossian [AT] work on Drinfel'd associators and KashiwaraVergne can be reinterpreted as a study of wknotted trivalent graphs.
The true value of wknots, 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 vknots. We expect vknotted objects to provide the global context whose projectivization (or "associated graded structure") will be the EtingofKazhdan theory of deformation quantization of Lie bialgebras [EK].
Related Mathematica Notebooks. "The Kishino Braid" (Source, PDF), "Dimensions" (Source, PDF), "wA" (Source, PDF), "InfinitesimalAlexanderModules" (Source, PDF).
Related talks. Oberwolfach0805, MSRI0808, Northeastern081028, Trieste0905, Bonn0908, Caen1206.
Links. SandersonsGarden.html.
Related Scratch Work is under Pensieve: WKO and Pensieve: Arrow_Diagrams_and_gl(N).
References.
[AT] ^ A. Alekseev and C. Torossian, The KashiwaraVergne conjecture and Drinfeld's associators, arXiv:0802.4300.
[BP] ^ B. Berceanu and S. Papadima, Universal Representations of Braid and BraidPermutation Groups, arXiv:0708.0634.
[BH] ^ T. Brendle and A. Hatcher, Configuration Spaces of Rings and Wickets, arXiv:0805.4354.
[EK] ^ P. Etingof and D. Kazhdan, Quantization of Lie Bialgebras, I, Selecta Mathematica, New Series 2 (1996) 141, arXiv:qalg/9506005.
[FRR] ^ R. Fenn, R. Rimanyi and C. Rourke, The BraidPermutation Group, Topology 36 (1997) 123135.
[Gol] ^ D. L. Goldsmith, The Theory of Motion Groups, Mich. Math. J. 281 (1981) 317.
[KV] ^ M. Kashiwara and M. Vergne, The CampbellHausdorff Formula and Invariant Hyperfunctions, Invent. Math. 47 (1978) 249272.
[Mc] ^ J. McCool, On BasisConjugating Automorphisms of Free Groups, Can. J. Math. 386(1986) 15251529.
[Sa] ^ S. Satoh, Virtual Knot Presentations of Ribbon Torus Knots, J. of Knot Theory and its Ramifications 94 (2000) 531542.