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Qingtao Chen

Publications and source records attributed to Qingtao Chen.

At least 19 recordsLinked to original sources
arXiv2025

On the asymptotic expansions of various quantum invariants I: the colored Jones polynomial of twist knots at the root of unity $e^{\frac{2π\sqrt{-1}}{N+\frac{1}{2}}}$

This is the first article in a series devoted to the study of the asymptotic expansions of various quantum invariants related to the twist knots. In this paper, by using the saddle point method developed by Ohtsuki, we obtain an asymptotic expansion formula for the colored Jones polynomial of twist knots $\mathcal{K}_p$ with $p\geq 6$ at the root of unity $e^{\frac{2π\sqrt{-1}}{N+\frac{1}{2}}}$.

math.GT↗
arXiv2024

On the asymptotic expansion of quantum invariants related to surgeries of Whitehead link I: Relative Reshetikhin-Turaev invariants and the Turaev-Viro invariants at $e^{\frac{2π\sqrt{-1}}{N+\frac{1}{2}}}$

In this article, we obtain an asymptotic expansion formula for the relative Reshetikhin-Turaev invariant in the case that the ambient 3-manifold is gained by doing rational surgery along one component of Whitehead link. In addition, we obtain an asymptotic expansion formula for the Turaev-Viro invariant of the cusped 3-manifold which is gained by doing rational surgery along one component of the Whitehead link.

math.GT↗
arXiv2024

On the asymptotic expansion of various quantum invariants III: the Reshetikhin-Turaev invariants of closed hyperbolic 3-manifolds obtained by doing integral surgery along the twist knot

This is the third article in a series devoted to the study of the asymptotic expansions of various quantum invariants related to the twist knots. In this paper, by using the saddle point method developed by Ohtsuki and Yokota, we obtain an asymptotic expansion formula for the Reshetikhin-Turaev invariants of closed hyperbolic 3-manifolds obtained by doing integral $q$-surgery along the twist knots $\mathcal{K}_p$ at the root of unity $e^{\frac{4π\sqrt{-1}}{r}}$ ($r$ is odd).

math.GT↗
arXiv2023

On the asymptotic expansions of various quantum invariants II: the colored Jones polynomial of twist knots at the root of unity $e^{\frac{2π\sqrt{-1}}{N+\frac{1}{M}}}$ and $e^{\frac{2π\sqrt{-1}}{N}}$

This is the second article in a series devoted to the study of the asymptotic expansions of various quantum invariants related to the twist knots. In this article, following the method and results in \cite{CZ23-1}, we present an asymptotic expansion formula for the colored Jones polynomial of twist knot $\mathcal{K}_p$ with $p\geq 6$ at the root of unity $e^{\frac{2π\sqrt{-1}}{N+\frac{1}{M}}}$ with $M\geq 2$. Furthermore, by taking the limit $M\rightarrow +\infty$, we obtain an asymptotic expansion formula for the colored Jones polynomial of twist knots $\mathcal{K}_p$ with $p\geq 6$ at the root of unity $e^{\frac{2π\sqrt{-1}}{N}}$.

math.GT↗
arXiv2021

Asymptotics of quantum $6j$ symbols

Asymptotics of quantum $6j$ symbols corresponding to a hyperbolic tetrahedra is investigated and the first two leading terms are determined for the case that the tetrahedron has a ideal or ultra-ideal vertex. These terms are given by the volume and the determinant of the Gram matrix of the tetrahedron. A relation to the volume conjecture of the Turaev-Viro invariant is also discussed.

math.QA↗
arXiv2019

Uni-traveling-carrier photodetector with high-contrast grating focusing-reflection mirrors

A novel uni-traveling-carrier photodetector (UTC-PD) structure with an integrated focusing-reflection (FR) mirror realized by a non-periodic concentric circular high-contrast grating (NP-CC-HCG), referred to as FR-UTC-PD, is proposed to enhance responsivity in conventional UTC-PDs. The FR-UTC-PD allows improving the responsivity by 36.5% at a 1.55-um wavelength as compared to a UTC-PD without integrated an FR mirror with 84.59% reflectivity. For 40-um-diameter PDs, the obtained 3-dB bandwidths are unaltered with values of 18 GHz at -3.0 V bias voltage. The radio-frequency (RF) output power and photocurrent are -1.77 dBm and 17.56 mA, respectively, at 10 GHz and the -6.0 V bias voltage.

physics.ins-det↗
arXiv2018

Volume conjectures for the Reshetikhin-Turaev and the Turaev-Viro invariants

We consider the asymptotics of the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic $3$-manifold, evaluated at the root of unity $\exp({2π\sqrt{-1}}/{r})$ instead of the standard $\exp({π\sqrt{-1}}/{r})$. We present evidence that, as $r$ tends to $\infty$, these invariants grow exponentially with growth rates respectively given by the hyperbolic and the complex volume of the manifold. This reveals an asymptotic behavior that is different from that of Witten's Asymptotic Expansion Conjecture, which predicts polynomial growth of these invariants when evaluated at the standard root of unity. This new phenomenon suggests that the Reshetikhin-Turaev invariants may have a geometric interpretation other than the original one via $SU(2)$ Chern-Simons gauge theory.

math.GT↗
arXiv2016

Cyclotomic expansion and volume conjecture for superpolynomials of colored HOMFLY-PT homology and colored Kauffman homology

We first study superpolynomial associated to triply-graded reduced colored HOMFLY-PT homology. We propose conjectures of congruent relations and cyclotomic expansion for it. We prove conjecture of $N=1$ for torus knot case, through which we obtain the corresponding invariant $α(T(m,n))=-(m-1)(n-1)/2$. This is closely related to the Milnor conjecture. Many examples including homologically thick knots and higher representations are also tested. Based on these examples, we further propose a conjecture that invariant $α$ determined in cyclotomic expansion at $N=1$ is a lower bound for smooth 4-ball genus. According to the structure of cyclotomic expansion, we propose a volume conjecture for $SU(n)$ specialized superpolynomial associated to reduced colored HOMFLY homology. We also prove the figure eight case for this new volume conjecture. Then we study superpolynomial associated to triply-graded reduced colored Kauffman homology. We propose a conjecture of cyclotomic expansion for it. Homologically thick examples and higher representations are tested. Finally we apply the same idea to the Heegaard-Floer knot homology and also obtain an expansion formula for all the examples we tested.

math.QA↗
arXiv2015

Congruent skein relations for colored HOMFLY-PT invariants and colored Jones polynomials

Colored HOMFLY-PT invariant, the generalization of the colored Jones polynomial, is one of the most important quantum invariants of links. This paper is devoted to investigating the basic structures of the colored HOMFLY-PT invariants of links. By using the HOMFLY-PT skein theory, firstly, we show that the (reformulated) colored HOMFLY-PT invariants actually lie in the ring $\mathbb{Z}[(q-q^{-1})^2,t^{\pm 1}]$. Secondly, we establish some symmetric formulas for colored HOMFLY-PT invariants of links, which include the rank-level duality as an easy consequence. Finally, motivated by the Labastida-Mariño-Ooguri-Vafa conjecture for framed links, we propose congruent skein relations for (reformulated) colored HOMFLY-PT invariants which are the generalizations of the skein relation for classical HOMFLY-PT polynomials. Then we study the congruent skein relation for colored Jones polynomials. In fact, we obtain a succinct formula for the case of knot. As an application, we prove a vanishing result for Reshetikhin-Turaev invariants of a family of 3-manifolds. Finally we study the congruent skein relations for $SU(n)$ quantum invariants.

math.GT↗
arXiv2015

Volume conjecture for $SU(n)$-invariants

This paper discuss an intrinsic relation among congruent relations \cite{CLPZ}, cyclotomic expansion and Volume Conjecture for $SU(n)$ invariants. Motivated by the congruent relations for $SU(n)$ invariants obtained in our previous work \cite{CLPZ}, we study certain limits of the $SU(n)$ invariants at various roots of unit. First, we prove a new symmetry property for the $SU(n)$ invariants by using a symmetry of colored HOMFLYPT invariants. Then we propose some conjectural formulas including the cyclotomic expansion conjecture and volume conjecture for $SU(n)$ invariants (specialization of colored HOMFLYPT invariants). We also give the proofs of these conjectural formulas for the case of figure-eight knot.

math.QA↗
arXiv2014

Full Colored HOMFLYPT Invariants, Composite Invariants and Congruent Skein Relation

In this paper, we investigate the properties of the full colored HOMFLYPT invariants in the full skein of the annulus $\mathcal{C}$. We show that the full colored HOMFLYPT invariant has a nice structure when $q\rightarrow 1$. The composite invariant is a combination of the full colored HOMFLYPT invariants. In order to study the framed LMOV type conjecture for composite invariants, we introduce the framed reformulated composite invariant $\check{\mathcal{R}}_{p}(\mathcal{L})$. By using the HOMFLY skein theory, we prove that $\check{\mathcal{R}}_{p}(\mathcal{L})$ lies in the ring $2\mathbb{Z}[(q-q^{-1})^2,t^{\pm 1}]$. Furthermore, we propose a conjecture of congruent skein relation for $\check{\mathcal{R}}_{p}(\mathcal{L})$ and prove it for certain special cases.

math.QA↗
arXiv2014

Recursion Formulas for HOMFLY and Kauffman Invariants

In this note we describe the recursion relations between two parameter HOMLFY and Kauffman polynomials of framed links These relation correspond to embeddings of quantized universal enveloping algebras. The relation corresponding to embeddings $g_{n}\supset g_{k}\times sl_{n-k}$ where $g_{n}$ is either $so_{2n+1}$, $so_{2n}$ or $sp_{2n}$ is new.

math.QA↗
arXiv2013

New structure for orthogonal quantum group invariants

Based on the orthogonal Labastida-Mari{ñ}o-Ooguri-Vafa conjecture made by L. Chen & Q. Chen [5], we derive an infinite product formula for Chern-Simons partition functions, which generalizes the Liu-Peng's [19] recent results to the orthogonal case. Symmetry property of this new infinite product structure is also discussed.

math.QA↗
arXiv2010

Orthogonal Quantum Group Invariants of Links

We study the Chern-Simons partition function of orthogonal quantum group invariants, and propose a new orthogonal Labastida-Mariño-Ooguri-Vafa conjecture as well as degree conjecture for free energy associated to the orthogonal Chern-Simons partition function. We prove the degree conjecture and some interesting cases of orthogonal LMOV conjecture. In particular, We provide a formula of colored Kauffman polynomials for torus knots and links, and applied this formula to verify certain case of the conjecture at roots of unity except $1$. We also derive formulas of Lickorish-Millett type for Kauffman polynomials and relate all these to the orthogonal LMOV conjecture.

math.QA↗
arXiv2010

Generalized Witten Genus and Vanishing Theorems

We construct a generalized Witten genus for spin$^c$ manifolds, which takes values in level 1 modular forms with integral Fourier expansion on a class of spin$^c$ manifolds called string$^c$ manifolds. We also construct a mod 2 analogue of the Witten genus for $8k+2$ dimensional spin manifolds. The Landweber-Stong type vanishing theorems are proven for the generalized Witten genus and the mod 2 Witten genus on string$^c$ and string (generalized) complete intersections in (product of) complex projective spaces respectively.

math.DG↗