On the asymptotic expansions of the Kashaev invariant of hyperbolic knots with seven crossings

2017 ◽  
Vol 28 (13) ◽  
pp. 1750096 ◽  
Author(s):  
Tomotada Ohtsuki
Keyword(s):  
Asymptotic Expansion ◽  
Saddle Point ◽  
Asymptotic Expansions ◽  
Technical Difficulty ◽  
Point Method ◽  
Saddle Point Method ◽  
Cambridge Philos ◽  
Volume Conjecture ◽  
Hyperbolic Volume ◽  
Chern Simons

We give presentations of the asymptotic expansions of the Kashaev invariant of hyperbolic knots with seven crossings. As the volume conjecture states, the leading terms of the expansions present the hyperbolic volume and the Chern–Simons invariant of the complements of the knots. As coefficients of the expansions, we obtain a series of new invariants of the knots. This paper is a continuation of the previous papers [T. Ohtsuki, On the asymptotic expansion of the Kashaev invariant of the [Formula: see text] knot, Quantum Topol. 7 (2016) 669–735; T. Ohtsuki and Y. Yokota, On the asymptotic expansion of the Kashaev invariant of the knots with 6 crossings, to appear in Math. Proc. Cambridge Philos. Soc.], where the asymptotic expansions of the Kashaev invariant are calculated for hyperbolic knots with five and six crossings. A technical difficulty of this paper is to use 4-variable saddle point method, whose concrete calculations are far more complicated than the previous papers.

On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings

2017 ◽  
Vol 165 (2) ◽  
pp. 287-339 ◽  
Author(s):  
TOMOTADA OHTSUKI ◽  
YOSHIYUKI YOKOTA
Keyword(s):  
Asymptotic Expansion ◽  
Saddle Point ◽  
Asymptotic Expansions ◽  
Saddle Point Method ◽  
Hyperbolic Structure ◽  
3 Dimensional ◽  
Volume Conjecture ◽  
Hyperbolic Volume ◽  
Chern Simons ◽  
Dimensional Domain

AbstractWe give presentations of the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings. In particular, we show the volume conjecture for these knots, which states that the leading terms of the expansions present the hyperbolic volume and the Chern--Simons invariant of the complements of the knots. As higher coefficients of the expansions, we obtain a new series of invariants of these knots.A non-trivial part of the proof is to apply the saddle point method to calculate the asymptotic expansion of an integral which presents the Kashaev invariant. A key step of this part is to give a concrete homotopy of the (real 3-dimensional) domain of the integral in ℂ3 in such a way that the boundary of the domain always stays in a certain domain in ℂ3 given by the potential function of the hyperbolic structure.

scholarly journals DIMENSIONAL TRANSMUTATION AND SYMMETRY BREAKING IN MAXWELL-CHERN-SIMONS SCALAR QED

1996 ◽  
Vol 11 (20) ◽  
pp. 1627-1635 ◽  
Author(s):  
F.S. NOGUEIRA ◽  
N.F. SVAITER
Keyword(s):  
Symmetry Breaking ◽  
Saddle Point ◽  
Effective Potential ◽  
Functional Integral ◽  
Point Method ◽  
Saddle Point Method ◽  
Integral Formalism ◽  
Dimensional Transmutation ◽  
Chern Simons ◽  
Functional Integral Formalism

The mechanism of dimensional transmutation is discussed in the context of Maxwell-Chern-Simons scalar QED. We evaluate the effective potential using the saddle point method through the functional integral formalism. An instability is found for λ>λ c where λ is the quartic scalar self-coupling. It is found that the symmetry breaking vacuum is more stable when the Chern-Simons mass is different from zero.

scholarly journals A Note on the Asymptotic Behavior of the Heights in $b$-Tries for $b$ Large

10.37236/1517 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Charles Knessl ◽  
Wojciech Szpankowski
Keyword(s):  
Asymptotic Behavior ◽  
Saddle Point ◽  
Generating Functions ◽  
Single Point ◽  
Extreme Value ◽  
Limiting Distribution ◽  
Point Method ◽  
Saddle Point Method ◽  
Numerical Verification ◽  
Height Distribution

We study the limiting distribution of the height in a generalized trie in which external nodes are capable to store up to $b$ items (the so called $b$-tries). We assume that such a tree is built from $n$ random strings (items) generated by an unbiased memoryless source. In this paper, we discuss the case when $b$ and $n$ are both large. We shall identify five regions of the height distribution that should be compared to three regions obtained for fixed $b$. We prove that for most $n$, the limiting distribution is concentrated at the single point $k_1=\lfloor \log_2 (n/b)\rfloor +1$ as $n,b\to \infty$. We observe that this is quite different than the height distribution for fixed $b$, in which case the limiting distribution is of an extreme value type concentrated around $(1+1/b)\log_2 n$. We derive our results by analytic methods, namely generating functions and the saddle point method. We also present some numerical verification of our results.

scholarly journals The Saddle Point Method for the Integral of the Absolute Value of the Brownian Motion

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Leonid Tolmatz
Keyword(s):  
Distribution Function ◽  
Brownian Motion ◽  
Saddle Point ◽  
Point Method ◽  
Saddle Point Method ◽  
Tail Asymptotics ◽  
Absolute Value ◽  
Exact Tail Asymptotics ◽  
International Audience ◽  
The Absolute

International audience The distribution function of the integral of the absolute value of the Brownian motion was expressed by L.Takács in the form of various series. In the present paper we determine the exact tail asymptotics of this distribution function. The proposed method is applicable to a variety of other Wiener functionals as well.

scholarly journals Sommes friables d'exponentielles et applications

2015 ◽  
Vol 67 (3) ◽  
pp. 597-638 ◽  
Author(s):  
Sary Drappeau
Keyword(s):  
Asymptotic Formula ◽  
Saddle Point ◽  
Exponential Sums ◽  
Prime Factor ◽  
Character Sums ◽  
Contour Integration ◽  
Point Method ◽  
Saddle Point Method ◽  
Greatest Prime Factor

AbstractAn integer is said to be y–friable if its greatest prime factor is less than y. In this paper, we obtain estimates for exponential sums over y–friable numbers up to x which are non–trivial when y ≥ . As a consequence, we obtain an asymptotic formula for the number of y-friable solutions to the equation a + b = c which is valid unconditionally under the same assumption. We use a contour integration argument based on the saddle point method, as developped in the context of friable numbers by Hildebrand and Tenenbaum, and used by Lagarias, Soundararajan and Harper to study exponential and character sums over friable numbers.

scholarly journals Power partitions and saddle-point method

2019 ◽  
Vol 204 ◽  
pp. 435-445 ◽  
Author(s):  
Gérald Tenenbaum ◽  
Jie Wu ◽  
Ya-Li Li
Keyword(s):  
Saddle Point ◽  
Point Method ◽  
Saddle Point Method
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