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

The Quality Control of Puerariae Lobatae Radix and Puerariae Thomsonii Radix

Puerariae Lobatae Radix (PLR) and Puerariae Thomsonii Radix (PTR) are traditional Chinese medicines used interchangeably in clinical practice, even though they possess significantly different chemical profiles. The aim of this thesis was to differentiate PLR from PTR using various analytical instruments coupled with chemometrics. Morphological results illustrate PLR possessed distinct macroscopic and microscopic features as compared to PTR. UPLC results reveal isoflavonoids were the major chemical constituents in both species, with the content of puerarin in PLR significantly greater than in PTR. PLS-DA models demonstrate both UPLC and HPTLC chromatographic fingerprints were effective in differentiating PLR from PTR. PLSR coupled with Raman spectra was able to predict the TPC and antioxidant capacities of PLR and PTR. The pharmacological results illustrate PLR possessed significantly greater anti-diabetic, cytoprotective and anti-cancer activities as compared to PTR. In summary, the results reveal the chemical fingerprints coupled with chemometrics was effective in differentiating PLR from PTR, and PLR was morphologically, chemically and pharmacologically different from PTR. This thesis provided further insight into the comprehensive nature of the quality control of two similar species and recommends changes to their descriptions in the pharmacopoeias. This will ultimately improve the quality, safety and efficacy of herbal products

Relative Reshetikhin-Turaev invariants, hyperbolic cone metrics and discrete Fourier transforms II

We prove the Volume Conjecture for the relative Reshetikhin-Turaev invariants proposed in [29] for all pairs (M,K) such that M\K is homeomorphic to the complement of the figure-8 knot in S^3 with almost all possible cone angles.Comment: 34 pages, 5 figures. arXiv admin note: substantial text overlap with arXiv:2003.1005

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

Relative Reshetikhin-Turaev invariants, hyperbolic cone metrics and discrete Fourier transforms I

We propose the Volume Conjecture for the relative Reshetikhin-Turaev invariants of a closed oriented $3$-manifold with a colored framed link inside it whose asymptotic behavior is related to the volume and the Chern-Simons invariant of the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring. We prove the conjecture in the case that the ambient $3$-manifold is obtained by doing an integral surgery along some components of a fundamental shadow link and the complement of the link in the ambient manifold is homeomorphic to the fundamental shadow link complement, for sufficiently small cone angles. Together with Costantino and Thurston's result that all compact oriented $3$-manifolds with toroidal or empty boundary can be obtained by doing an integral surgery along some components of a suitable fundamental shadow link, this provides a possible approach of solving Chen-Yang's Volume Conjecture for the Reshetikhin-Turaev invariants of closed oriented hyperbolic $3$-manifolds. We also introduce a family of topological operations (the change of pair operations) that connect all pairs of a closed oriented $3$-manifold and a framed link inside it that have homeomorphic complements, which correspond to doing the partial discrete Fourier transforms to the corresponding relative Reshetikhin-Turaev invariants. As an application, we find a Poisson Summation Formula for the discrete Fourier transforms.Comment: 44 pages, 21 figure

Generalized Bonahon-Wong-Yang volume conjecture of quantum invariants of surface diffeomorphisms I: the figure eight knot complement

We propose a generalization of the Bonahon-Wong-Yang volume conjecture of quantum invariants of surface diffeomorphisms, by relating the asymptotics of the invariants with certain hyperbolic cone structure on the mapping torus determined by the choice of the invariant puncture weights. We prove the conjecture for the once-punctured torus bundle with the diffeomorphism given by the word $LR$.Comment: 29 pages, 1 figur