WKO
The information below is preliminary and cannot be trusted! (v)
Finite Type Invariants of W-Knotted Objects: From Alexander to Kashiwara and Vergne
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.
The group of w-braids 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 "basis-conjugating" automorphisms of a free group [math]\displaystyle{ F_n }[/math] - the smallest subgroup of [math]\displaystyle{ \operatorname{Aut}(F_n) }[/math] 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 [math]\displaystyle{ {\mathbb R}^3 }[/math]. Satoh [Sa] studied several classes of w-knotted objects (under the name "weakly-virtual") and has shown them to be closely related to certain classes of knotted surfaces in [math]\displaystyle{ {\mathbb R}^4 }[/math]. So w-knotted objects are algebraically and topologically interesting.
In this article we study finite type invariants of several classes of w-knotted objects. Following Berceanu and Papadima [BP], we construct a homomorphic universal finite type invariant of w-braids, and hence show that the McCool group of automorphisms is "1-formal". We also construct a homomorphic universal finite type invariant of w-tangles. We find that the universal finite type invariant of w-knots is more or less the Alexander polynomial (details inside).
Much as the spaces [math]\displaystyle{ {\mathcal A} }[/math] of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, we find that the spaces [math]\displaystyle{ {\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 [KV] conjecture and much of the Alekseev-Torrosian [AT] work on Drinfel'd associators and Kashiwara-Vergne can be re-interpreted as a study of w-knotted trivalent graphs.
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 [EK].
The paper. WKO.pdf, WKO.zip.
Related Mathematica Notebooks. "The Kishino Braid" (Source, PDF), "Dimensions" (Source, PDF), "wA" (Source, PDF), "InfinitesimalAlexanderModules" (Source, PDF).
Related talks. Oberwolfach-0805, MSRI-0808, Northeastern-081028, Trieste-0905, Bonn-0908.
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 Kashiwara-Vergne conjecture and Drinfeld's associators, arXiv:0802.4300.
[BP] ^ B. Berceanu and S. Papadima, Universal Representations of Braid and Braid-Permutation 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) 1-41, arXiv:q-alg/9506005.
[FRR] ^ R. Fenn, R. Rimanyi and C. Rourke, The Braid-Permutation Group, Topology 36 (1997) 123-135.
[Gol] ^ D. L. Goldsmith, The Theory of Motion Groups, Mich. Math. J. 28-1 (1981) 3-17.
[KV] ^ M. Kashiwara and M. Vergne, The Campbell-Hausdorff Formula and Invariant Hyperfunctions, Invent. Math. 47 (1978) 249-272.
[Mc] ^ J. McCool, On Basis-Conjugating Automorphisms of Free Groups, Can. J. Math. 38-6(1986) 1525-1529.
[Sa] ^ S. Satoh, Virtual Knot Presentations of Ribbon Torus Knots, J. of Knot Theory and its Ramifications 9-4 (2000) 531-542.
