On the Combinatorial Structure of Primitive Vassiliev Invariants, III - A Lower Bound

We prove that the dimension of the space of primitive Vassiliev invariants of degree n grows - as n tends to infinity - faster than Exp(c Sqrt(n)) for any c < Pi Sqrt (2/3). The proof relies on the use of the weight systems coming from the Lie algebra gl(N). In fact, we show that our bound is - up to multiplication with a rational function in n - the best possible that one can get with gl(N)-weight systems.Comment: 11 pages, 12 figure

Extremal Khovanov homology of Turaev genus one links

The Turaev genus of a link can be thought of as a way of measuring how non-alternating a link is. A link is Turaev genus zero if and only if it is alternating, and in this viewpoint, links with large Turaev genus are very non-alternating. In this paper, we study Turaev genus one links, a class of links which includes almost alternating links. We prove that the Khovanov homology of a Turaev genus one link is isomorphic to $\mathbb{Z}$ in at least one of its extremal quantum gradings. As an application, we compute or nearly compute the maximal Thurston Bennequin number of a Turaev genus one link.Comment: 30 pages, 18 figure

On the Head and the Tail of the Colored Jones Polynomial

The colored Jones polynomial is a series of one variable Laurent polynomials J(K,n) associated with a knot K in 3-space. We will show that for an alternating knot K the absolute values of the first and the last three leading coefficients of J(K,n) are independent of n when n is sufficiently large. Computation of sample knots indicates that this should be true for any fixed leading coefficient of the colored Jones polynomial for alternating knots. As a corollary we get a Volume-ish Theorem for the colored Jones Polynomial.Comment: 14 pages, 6 figure