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

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 $M$-th colored Jones polynomials evaluated at $(N+1/2)$-th root of unity with a fixed limiting ratio, $s$, of $M$ and $(N+1/2)$. We find out the asymptotic expansion formula (AEF) of the colored Jones polynomials of the figure eight knot with $s$ close to $1$. Nonetheless, we show that the exponential growth rate of the colored Jones polynomials of the figure eight knot with $s$ close to $1/2$ is strictly less than those with $s$ close to $1$. 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 $s$ 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

Geometry of fundamental shadow link complements and applications to the 1-loop conjecture

We construct a geometric ideal triangulation for every fundamental shadow link complement and solve the gluing equation explicitly in terms of the holonomies of the meridians of the link for any generic character in the distinguished component of the $\mathrm{PSL}(2;\mathbb{C})$-character variety of the link complement. As an application, we obtain a new formula for the volume of a hyperideal tetrahedron in terms of its dihedral angles. Moreover, by using the ideal triangulation, we verify the 1-loop conjecture proposed by Dimofte and Garoufalidis for every fundamental shadow link complement. We also prove a surgery formula for the 1-loop invariant with respect to certain nice ideal triangulations of 3-manifolds with toroidal boundary.Comment: 52 pages, 22 figure

Mainland Chinese Tourists’ Expectations, Perceived Performance of and Satisfaction towards Shopping Malls in Hong Kong

Tourists from mainland China constitute one of the world's biggest and fastest-growing travel markets. The Hong Kong Tourism Board expects the frequency of the mainland Chinese tourists to grow steadily. One of the beneficiaries of this phenomenon is Hong Kong’s shopping malls. However, understanding of tourists from mainland China’s expectations from, perceived performance of and satisfaction with the shopping malls’ attributes in Hong Kong is inadequate. This study intends to fill this important gap. A survey questionnaire was employed for data collection. The main results reveal the mainland Chinese tourists’ levels of satisfaction, indifference and dissatisfaction as well as the relationship between shopping malls’ attributes and customer satisfaction