20 research outputs found

    The Dynamics of Off-center Reflection

    Get PDF
    The dynamical properties, especially the symmetric orbits, of the 2-parameter family of circle maps called off-center reflection is studied.Comment: 15 pages, 11 figures, AMS-LaTeX, revised version, no offset proble

    Asymptotic Behavior of Colored Jones polynomial and Turaev-Viro Invariant of figure eight knot

    Full text link
    In this paper we investigate the asymptotic behavior of the colored Jones polynomials and the Turaev-Viro invariants for the figure eight knot. More precisely, we consider the MM-th colored Jones polynomials evaluated at (N+1/2)(N+1/2)-th root of unity with a fixed limiting ratio, ss, of MM and (N+1/2)(N+1/2). We find out the asymptotic expansion formula (AEF) of the colored Jones polynomials of the figure eight knot with ss close to 11. Nonetheless, we show that the exponential growth rate of the colored Jones polynomials of the figure eight knot with ss close to 1/21/2 is strictly less than those with ss close to 11. It is known that the Turaev Viro invariant of the figure eight knot can be expressed in terms of a sum of its colored Jones polynomials. Our results show that this sum is asymptotically equal to the sum of the terms with ss close to 1. As an application of the asymptotic behavior of the colored Jones polynomials, we obtain the asymptotic expansion formula for the Turaev-Viro invariants of the figure eight knot. Finally, we suggest a possible generalization of our approach so as to relate the AEF for the colored Jones polynomials and the AEF for the Turaev-Viro invariants for general hyperbolic knots.Comment: 40 pages, 0 figure
    corecore