Two applications of elementary knot theory to Lie algebras and Vassiliev invariants

Bar-Natan, Dror¹ ; Le, Thang T Q² ; Thurston, Dylan P³

¹ Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 3G3, Canada
² Department of Mathematics, SUNY at Buffalo, Buffalo, New York 14214, USA
³ Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138, USA

Geometry & topology, Tome 7 (2003) no. 1, pp. 1-31.

Résumé

Using elementary equalities between various cables of the unknot and the Hopf link, we prove the Wheels and Wheeling conjectures, which give, respectively, the exact Kontsevich integral of the unknot and a map intertwining two natural products on a space of diagrams. It turns out that the Wheeling map is given by the Kontsevich integral of a cut Hopf link (a bead on a wire), and its intertwining property is analogous to the computation of $1+1=2$ on an abacus. The Wheels conjecture is proved from the fact that the $k$–fold connected cover of the unknot is the unknot for all $k$.

Along the way, we find a formula for the invariant of the general $\left(k,l\right)$ cable of a knot. Our results can also be interpreted as a new proof of the multiplicativity of the Duflo–Kirillov map $S\left(\mathfrak{g}\right)\to U\left(\mathfrak{g}\right)$ for metrized Lie (super-)algebras $\mathfrak{g}$.

Détail

DOI : 10.2140/gt.2003.7.1

Keywords: Wheels, Wheeling, Vassiliev invariants, Hopf link, $1+1=2$, Duflo isomorphism, cabling

Bibliographie

[1] A Alekseev, E Meinrenken, The non-commutative Weil algebra, Invent. Math. 139 (2000) 135

[2] M Andler, A Dvorsky, S Sahi, Kontsevich quantization and invariant distributions on Lie groups, Ann. Sci. École Norm. Sup. $(4)$ 35 (2002) 371

[3] D Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423

[4] D Bar-Natan, Non-associative tangles, from: "Geometric topology (Athens, GA, 1993)", AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 139

[5] D Bar-Natan, S Garoufalidis, L Rozansky, D P Thurston, Wheels, wheeling, and the Kontsevich integral of the unknot, Israel J. Math. 119 (2000) 217

[6] D Bar-Natan, S Garoufalidis, L Rozansky, D P Thurston, The \AArhus integral of rational homology 3–spheres II: Invariance and universality, Selecta Math. $($N.S.$)$ 8 (2002) 341

[7] D Bar-Natan, R Lawrence, A rational surgery formula for the LMO invariant, Israel J. Math. 140 (2004) 29

[8] S V Chmutov, A N Varchenko, Remarks on the Vassiliev knot invariants coming from $\mathrm{ sl}_2$, Topology 36 (1997) 153

[9] D Bar-Natan, Vassiliev and quantum invariants of braids, from: "The interface of knots and physics (San Francisco, CA, 1995)", Proc. Sympos. Appl. Math. 51, Amer. Math. Soc. (1996) 129

[10] J Dixmier, Enveloping algebras, Graduate Studies in Mathematics 11, American Mathematical Society (1996)

[11] M Duflo, Opérateurs différentiels bi-invariants sur un groupe de Lie, Ann. Sci. École Norm. Sup. $(4)$ 10 (1977) 265

[12] R Godement, Introduction à la théorie des groupes de Lie, Springer (2004)

[13] V Hinich, A Vaintrob, Cyclic operads and algebra of chord diagrams, Selecta Math. $($N.S.$)$ 8 (2002) 237

[14] N Hitchin, J Sawon, Curvature and characteristic numbers of hyper–Kähler manifolds, Duke Math. J. 106 (2001) 599

[15] M Kontsevich, Vassiliev's knot invariants, from: "I. M. Gel'fand Seminar", Adv. Soviet Math. 16, Amer. Math. Soc. (1993) 137

[16] M Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003) 157

[17] A Kricker, B Spence, I Aitchison, Cabling the Vassiliev invariants, J. Knot Theory Ramifications 6 (1997) 327

[18] T Q T Le, J Murakami, Representation of the category of tangles by Kontsevich's iterated integral, Comm. Math. Phys. 168 (1995) 535

[19] T Q T Le, J Murakami, The universal Vassiliev–Kontsevich invariant for framed oriented links, Compositio Math. 102 (1996) 41

[20] T T Q Le, J Murakami, Parallel version of the universal Vassiliev–Kontsevich invariant, J. Pure Appl. Algebra 121 (1997) 271

[21] T T Q Le, J Murakami, T Ohtsuki, On a universal perturbative invariant of 3–manifolds, Topology 37 (1998) 539

[22] J Lieberum, The number of independent Vassiliev invariants in the Homfly and Kauffman polynomials, Doc. Math. 5 (2000) 275

[23] T Mochizuki, On the morphism of Duflo–Kirillov type, J. Geom. Phys. 41 (2002) 73

[24] J Murakami, T Ohtsuki, Topological quantum field theory for the universal quantum invariant, Comm. Math. Phys. 188 (1997) 501

[25] D Thurston, Torus actions for the LMO invariant, in preparation

[26] P Vogel, Algebraic structures on modules of diagrams, technical report, Université Paris VII (1995)