52 research outputs found
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 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
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