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.

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.

Hyperasymptotics for integrals with saddles

1991 ◽  
Vol 434 (1892) ◽  
pp. 657-675 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Saddle Point ◽  
Multiple Scattering ◽  
Positive Constant ◽  
Asymptotic Expansions ◽  
Steepest Descent ◽  
The Other ◽  
Large Parameter ◽  
Original Series ◽  
Method Of Steepest Descent

Integrals involving exp { – k f ( z )}, where | k | is a large parameter and the contour passes through a saddle of f ( z ), are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles, through which the contour does not pass. These contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent. The refinement is achieved by means of an exact ‘resurgence relation', expressing the original integral as its truncated saddle-point asymptotic expansion plus a remainder involving the integrals through certain ‘adjacent’ saddles, determined by a topological rule. Iteration of the resurgence relation, and choice of truncation near the least term of the original series, leads to a representation of the integral as a sum of contributions associated with ‘multiple scattering paths’ among the saddles. No resummation of divergent series is involved. Each path gives a ‘hyperseries’, depending on the terms in the asymptotic expansions for each saddle (these depend on the particular integral being studied and so are non-universal), and certain ‘hyperterminant’ functions defined by integrals (these are always the same and hence universal). Successive hyperseries get shorter, so the scheme naturally halts. For two saddles, the ultimate error is approximately ∊ 2.386 , where ∊ (proportional to exp (— A │ k │) where A is a positive constant), is the error in optimal truncation of the original series. As a numerical example, an integral with three saddles is computed hyperasymptotically.

scholarly journals ASYMPTOTICS OF THE QUANTUM INVARIANTS FOR SURGERIES ON THE FIGURE 8 KNOT

2006 ◽  
Vol 15 (04) ◽  
pp. 479-548 ◽  
Author(s):  
JØRGEN ELLEGAARD ANDERSEN ◽  
SØREN KOLD HANSEN
Keyword(s):  
Jones Polynomial ◽  
Saddle Point Method ◽  
Stationary Points ◽  
Quantum Invariants ◽  
Contour Integrals ◽  
Volume Conjecture ◽  
Phase Functions ◽  
Chern Simons ◽  
Large Quantum ◽  
Double Contour

We investigate the Reshetikhin–Turaev invariants associated to SU(2) for the 3-manifolds M obtained by doing any rational surgery along the figure 8 knot. In particular, we express these invariants in terms of certain complex double contour integrals. These integral formulae allow us to propose a formula for the leading asymptotics of the invariants in the limit of large quantum level. We analyze this expression using the saddle point method. We construct a certain surjection from the set of stationary points for the relevant phase functions onto the space of conjugacy classes of nonabelian SL(2, ℂ)-representations of the fundamental group of M and prove that the values of these phase functions at the relevant stationary points equals the classical Chern–Simons invariants of the corresponding flat SU(2)-connections. Our findings are in agreement with the asymptotic expansion conjecture. Moreover, we calculate the leading asymptotics of the colored Jones polynomial of the figure 8 knot following Kashaev [14]. This leads to a slightly finer asymptotic description of the invariant than predicted by the volume conjecture [24].

LINK COMPLEMENTS ARISING FROM ARITHMETIC GROUP ACTIONS

1995 ◽  
Vol 06 (03) ◽  
pp. 337-370 ◽  
Author(s):  
FRITZ GRUNEWALD ◽  
ULRICH HIRSCH
Keyword(s):  
Quotient Space ◽  
Free Subgroup ◽  
Arithmetic Group ◽  
Hyperbolic Structure ◽  
3 Dimensional ◽  
Ring Of Integers ◽  
Hyperbolic Volume ◽  
Link Complements ◽  
Torsion Free Subgroup ◽  
Torsion Free

Let [Formula: see text] be a torsion-free subgroup acting discontinuously on 3-dimensional hyperbolic space [Formula: see text]. Assume further that Γ\ℍ3 has finite hyperbolic volume. The quotient-space Γ\ℍ3 is then a 3-manifold which can be compactified by the addition of finitely many 2-tori. This paper discusses a procedure which decides whether Γ\ℍ3 is homeomorphic to the complement of a link in S3. We apply our procedure to subgroups of low index in [Formula: see text], where [Formula: see text] is the ring of integers in [Formula: see text]. As a result we find new link complements having a complete hyperbolic structure coming from an arithmetic group. Finally we prove that up to conjugacy there are only finitely many commensurability classes of arithmetic subgroups [Formula: see text] so that Γ\ℍ3 is homeomorphic to the complement of a link in S3.

On the stability of a columnar vortex to disturbances with large azimuthal wavenumber: the lower neutral points

1987 ◽  
Vol 178 ◽  
pp. 549-566 ◽  
Author(s):  
K. Stewartson ◽  
S. Leibovich
Keyword(s):  
Asymptotic Expansion ◽  
Saddle Point ◽  
Saddle Point Method ◽  
Marginal Stability ◽  
Columnar Vortex ◽  
The Stability ◽  
Neutral Conditions ◽  
Inviscid Instability ◽  
Limiting Values ◽  
Swirl Parameter

The inviscid instability of a columnar trailing-line vortex at large values of the azimuthal wavenumber n near neutral conditions is considered. This extends an earlier analysis (Leibovich & Stewartson 1983), which is not accurate near the limiting values of the axial wavenumber for which instabilities exist. Here an asymptotic expansion is derived for the solution in the neighbourhood of the lower neutral point and the results compared with existing computations a t moderate values of n. For these weak instabilities disturbances are centred near the axis of the vortex and the relevant equation is solved in the complex plane by a generalized saddle-point method. In addition, the marginal stability of the vortex is examined, and an estimate obtained of the value of the swirl parameter above which the vortex is stable at large values of n.

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 BPS invariants for 3-manifolds at rational level K

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Hee-Joong Chung
Keyword(s):  
Asymptotic Expansion ◽  
Exact Expression ◽  
Roots Of Unity ◽  
Partition Functions ◽  
Knot Invariants ◽  
Volume Conjecture ◽  
Seifert Manifolds ◽  
Coprime Integers ◽  
Bps Invariants ◽  
Chern Simons

Abstract We consider the Witten-Reshetikhin-Turaev invariants or Chern-Simons partition functions at or around roots of unity $$ q={e}^{\frac{2\pi i}{K}} $$ q = e 2 πi K with a rational level K = $$ \frac{r}{s} $$ r s where r and s are coprime integers. From the exact expression for the G = SU(2) Witten-Reshetikhin-Turaev invariants of the Seifert manifolds at a rational level obtained by Lawrence and Rozansky, we provide an expected form of the structure of the Witten-Reshetikhin-Turaev invariants in terms of the homological blocks at a rational level. Also, we discuss the asymptotic expansion of knot invariants around roots of unity where we take a limit different from the limit in the standard volume conjecture.

scholarly journals Prime geodesics and averages of the Zagier L-series

2021 ◽  
pp. 1-24
Author(s):  
OLGA BALKANOVA ◽  
DMITRY FROLENKOV ◽  
MORTEN S. RISAGER
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Error Term ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Prime Geodesic Theorem ◽  
Error Terms ◽  
Average Size ◽  
Real Quadratic

Abstract The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.

Asymptotic Expansions in the Exponent: a Compound Poisson Approach

1997 ◽  
Vol 29 (02) ◽  
pp. 374-387 ◽  
Author(s):  
V. Čekanavičius
Keyword(s):  
Asymptotic Expansion ◽  
Large Deviations ◽  
Poisson Distribution ◽  
Asymptotic Expansions ◽  
Compound Poisson ◽  
Poisson Measures ◽  
Poisson Approximations

The accuracy of the Normal or Poisson approximations can be significantly improved by adding part of an asymptotic expansion in the exponent. The signed-compound-Poisson measures obtained in this manner can be of the same structure as the Poisson distribution. For large deviations we prove that signed-compound-Poisson measures enlarge the zone of equivalence for tails.

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