THE COLORED JONES POLYNOMIALS OF DOUBLES OF KNOTS

2008 ◽  
Vol 17 (08) ◽  
pp. 925-937
Author(s):  
TOSHIFUMI TANAKA
Keyword(s):  
Jones Polynomial ◽  
Colored Jones Polynomial ◽  
Volume Conjecture ◽  
Jones Polynomials ◽  
Skein Theory ◽  
Colored Jones Polynomials

We give formulas for the N-colored Jones polynomials of doubles of knots by using skein theory. As a corollary, we show that if the volume conjecture for untwisted positive (or negative) doubles of knots is true, then the colored Jones polynomial detects the unknot.

scholarly journals Stability properties of the colored Jones polynomial

2019 ◽  
Vol 28 (08) ◽  
pp. 1950050
Author(s):  
Christine Ruey Shan Lee
Keyword(s):  
Power Series ◽  
Jones Polynomial ◽  
Topological Information ◽  
Colored Jones Polynomial ◽  
Stability Properties ◽  
Link Complement

It is known that the colored Jones polynomial of a [Formula: see text]-adequate link has a well-defined tail consisting of stable coefficients, and that the coefficients of the tail carry geometric and topological information on the [Formula: see text]-adequate link complement. We show that a power series similar to the tail of the colored Jones polynomial for [Formula: see text]-adequate links can be defined for all links, and that it is trivial if and only if the link is non [Formula: see text]-adequate.

scholarly journals Normal modes in thermal AdS via the Selberg zeta function

2020 ◽  
Vol 9 (1) ◽  
Author(s):  
Victoria Martin ◽  
Andrew Svesko
Keyword(s):  
Zeta Function ◽  
Heat Kernel ◽  
Normal Mode ◽  
Kernel Method ◽  
Normal Modes ◽  
Quasinormal Mode ◽  
Partition Functions ◽  
Selberg Zeta Function ◽  
Exact Agreement ◽  
Spin Fields

The heat kernel and quasinormal mode methods of computing 1-loop partition functions of spin ss fields on hyperbolic quotient spacetimes \mathbb{H}^{3}/\mathbb{Z}ℍ3/ℤ are related via the Selberg zeta function. We extend that analysis to thermal \text{AdS}_{2n+1}AdS2n+1 backgrounds, with quotient structure \mathbb{H}^{2n+1}/\mathbb{Z}ℍ2n+1/ℤ. Specifically, we demonstrate the zeros of the Selberg function encode the normal mode frequencies of spin fields upon removal of non-square-integrable modes. With this information we construct the 1-loop partition functions for symmetric transverse traceless tensors in terms of the Selberg zeta function and find exact agreement with the heat kernel method.

scholarly journals Second variation of Selberg zeta functions and curvature asymptotics

2019 ◽  
Vol 57 (1) ◽  
pp. 23-60
Author(s):  
Ksenia Fedosova ◽  
Julie Rowlett ◽  
Genkai Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Vector Bundle ◽  
Zeta Function ◽  
Explicit Formula ◽  
Zeta Functions ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Second Variation ◽  
Selberg Zeta Function ◽  
The Second Variation

Abstract We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, Z(s), on Teichmüller space. We then use this formula to determine the asymptotic behavior as $$\mathfrak {R}s \rightarrow \infty $$Rs→∞ of the second variation. As a consequence, for $$m \in {\mathbb {N}}$$m∈N, we obtain the complete expansion in m of the curvature of the vector bundle $$H^0(X_t, {\mathcal {K}}_t)\rightarrow t\in {\mathcal {T}}$$H0(Xt,Kt)→t∈T of holomorphic m-differentials over the Teichmüller space $${\mathcal {T}}$$T, for m large. Moreover, we show that this curvature agrees with the Quillen curvature up to a term of exponential decay, $$O(m^2 \mathrm{e}^{-l_0 m}),$$O(m2e-l0m), where $$l_0$$l0 is the length of the shortest closed hyperbolic geodesic.

Sign in / Sign up

Export Citation Format

Share Document