384 research outputs found
A volume-ish theorem for the Jones polynomial of alternating knots
The Volume conjecture claims that the hyperbolic Volume of a knot is
determined by the colored Jones polynomial.
The purpose of this article is to show a Volume-ish theorem for alternating
knots in terms of the Jones polynomial, rather than the colored Jones
polynomial: The ratio of the Volume and certain sums of coefficients of the
Jones polynomial is bounded from above and from below by constants.
Furthermore, we give experimental data on the relation of the growths of the
hyperbolic volume and the coefficients of the Jones polynomial, both for
alternating and non-alternating knots.Comment: 14 page
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
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