Dror Bar-Natan

On the Vassiliev knot invariants

The theory of knot invariants of finite type (Vassiliev invariants) is described. These invariants turn out to be at least as powerful as the Jones polynomial and its numerous generalizations coming from various quantum groups, and it is conjectured that these invariants are precisely as powerful as those polynomials. As invariants of finite type are much easier to define and manipulate than the quantum group invariants, it is likely that in attempting to classify knots, invariants of finite type will play a more fundamental role than the various knot polynomials.

VASSILIEV HOMOTOPY STRING LINK INVARIANTS

We investigate Vassiliev homotopy invariants of string links, and find that in this particular case, most of the questions left unanswered in [3] can be answered affirmatively, In particular, Vassiliev invariants classify string links up to homotopy, and all Vassiliev homotopy string link invariants come from marked surfaces as in [3], using the same construction that in the case of knots gives the HOMFLY and Kauffman polynomials. In addition, the Milnor μ invariants of string links are shown to be Vassiliev invariants, and it is re-proven, by elementary means, that Vassiliev invariants classify braids.

PERTURBATIVE CHERN-SIMONS THEORY

We present the perturbation theory of the Chern-Simons gauge field theory and prove that to second order it indeed gives knot invariants. We identify these invariants and show that in fact we get a previously unknown integral formula for the Arf invariant of a knot, in complete agreement with earlier non-perturbative results of Witten. We outline our expectations for the behavior of the theory beyond two loops.

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

Using elementary equalities between various cables of the unknot and the Hopf link, we prove the Wheels and Wheeling conjectures of [5, 9], 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 nd a formula for the invariant of the general (k;l) cable of a knot. Our results can also be interpreted as a new proof of the multiplicativity of the Duflo{Kirillov map S(g)! U (g) for metrized Lie (super-)algebras g. AMS Classication numbers Primary: 57M27 Secondary: 17B20, 17B37

Wheels, wheeling, and the Kontsevich integral of the Unknot

We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne [De]) for the relation between the two natural products on the space of uni-trivalent diagrams. The two formulas use the related notions of “Wheels” and “Wheeing”. We prove these formulas ‘on the level of Lie algebras’ using standard techniques from the theory of Vassiliev invariants and the theory of Lie algebras. In a brief epilogue we report on recent proofs of our full conjectures, by Kontsevich [Ko2] and by DBN, DPT, and T. Q. T. Le, [BLT].

The Karoubi envelope and Lee's degeneration of Khovanov homology

We give a simple proof of Lee’s result from [5], that the dimension of the Lee variant of the Khovanov homology of a c ‐component link is 2 c , regardless of the number of crossings. Our method of proof is entirely local and hence we can state a Lee-type theorem for tangles as well as for knots and links. Our main tool is the “Karoubi envelope of the cobordism category”, a certain enlargement of the cobordism category which is mild enough so that no information is lost yet strong enough to allow for some simplifications that are otherwise unavailable. 57M25; 57M27, 18E05

A RATIONAL SURGERY FORMULA FOR THE LMO INVARIANT

We write a formula for the LMO invariant of a rational homology sphere presented as a rational surgery on a link inS3. Our main tool is a careful use of the Århus integral and the (now proven) “Wheels” and “Wheeling” conjectures of B-N, Garoufalidis, Rozansky and Thurston. As steps, side benefits and asides we give explicit formulas for the values of the Kontsevich integral on the Hopf link and on Hopf chains, and for the LMO invariant of lens spaces and Seifert fibered spaces. We find that the LMO invariant does not separate lens spaces, is far from separating general Seifert fibered spaces, but does separate Seifert fibered spaces which are integral homology spheres.

Lie algebras and the Four Color Theorem

We present a statement about Lie algebras that is equivalent to the Four Color Theorem.

Finite-type invariants of w-knotted objects, I: w-knots and the Alexander polynomial

This is the first in a series of papers studying w-knots, and more generally, w-knotted objects (w-braids, w-tangles, etc). These are classes of knotted objects which are wider, but weaker than their “usual” counterparts. The group of w-braids was studied (under the name “welded braids”) by Fenn, Rimanyi and Rourke and was shown to be isomorphic to the McCool group of “basisconjugating” automorphisms of a free group Fn : the smallest subgroup of Aut.Fn/ that contains both braids and permutations. Brendle and Hatcher, in work that traces back to Goldsmith, have shown this group to be a group of movies of flying rings in R 3 . Satoh studied several classes of w-knotted objects (under the name “weaklyvirtual”) and has shown them to be closely related to certain classes of knotted surfaces in R 4 . So w-knotted objects are algebraically and topologically interesting. Here we study finite-type invariants of w-braids and w-knots. Following Berceanu and Papadima, we construct homomorphic universal finite-type invariants of w-braids. The universal finite-type invariant of w-knots is essentially the Alexander polynomial. Much as the spaces A of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, the spaces A w 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. Later in this paper series we re-interpret the work of Alekseev and Torossian on Drinfel’d associators and the Kashiwara‐Vergne problem as a study of w-knotted trivalent graphs. 57M25, 57Q45

Some Dimensions of Spaces of Finite Type Invariants of Virtual Knots

We compute many dimensions of spaces of finite type invariants of virtual knots (of several kinds) and the dimensions of the corresponding spaces of “weight systems,” finding everything to be in agreement with the conjecture that “every weight system integrates.”