Narrow Results

We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of a cable of the figure-eight knot, evaluated at $\mathrm{\text{exp}}(\xi /N)$ for a real number $\xi$. We show that if $\xi$ is sufficiently large, the colored Jones polynomial grows exponentially when $N$ goes to the infinity. Moreover the growth rate is related to the Chern–Simons invariant of the knot exterior associated with an $\mathrm{\text{SL}}(2;ℝ)$ representation.

We study the asymptotic behaviors of the colored Jones polynomials of torus knots. Contrary to the works by R. Kashaev, O. Tirkkonen, Y. Yokota, and the author, they do not seem to give the volumes or the Chern–Simons invariants of the three-manifolds obtained by Dehn surgeries. On the other hand it is proved that in some cases the limits give the inverse of the Alexander polynomial.

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 $5_2$ 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.

In this paper, we study the variation of the Turaev–Viro invariants for $3$-manifolds with toroidal boundary under the operation of attaching a $(p,q)$-cable space. We apply our results to a conjecture of Chen and Yang which relates the asymptotics of the Turaev–Viro invariants to the simplicial volume of a compact oriented $3$-manifold. For $p$ and $q$ coprime, we show that the Chen–Yang volume conjecture is stable under $(p,q)$-cabling. We achieve our results by studying the linear operator ${\mathrm{\text{RT}}}_r$ associated to the torus knot cable spaces by the Reshetikhin–Turaev ${\mathrm{\text{SO}}}_3$-Topological Quantum Field Theory (TQFT), where the TQFT is well-known to be closely related to the desired Turaev–Viro invariants. In particular, our utilized method relies on the invertibility of the linear operator for which we provide necessary and sufficient conditions.

In this paper, we discuss the relation between the colored Jones polynomial of a 2-bridge link and the ideal triangulation of it's complement in S³. The aim of this paper is to describe the ideal triangulation of a 2-bridge link complement and to show that the hyperbolicity equations coincide with the equations obtained from the colored Jones polynomial of a 2-bridge link, and to compare this triangulation with the canonical decomposition of the 2-bridge link complement introduced by Sakuma and Weeks in [10].

We calculate certain limits of the colored Alexander invariant of the figure-eight knot for some cases and show that this limit is related to the volume of hyperbolic orbifolds whose singular set is the figure-eight knot.

For a hyperbolic knot, an ideal triangulation of the knot complement corresponding to the colored Jones polynomial was introduced by Thurston. Considering this triangulation of a twist knot, we find a function which gives the hyperbolicity equations and the complex volume of the knot complement, using Zickert's theory of the extended Bloch group and the complex volume.

We also consider a formal approximation of the colored Jones polynomial. Following Ohnuki's theory of 2-bridge knots, we define another function which comes from the approximation. We show that this function is essentially the same as the previous function, and therefore it also gives the same hyperbolicity equations and the complex volume.

Finally we compare this result with our previous one which dealt with Yokota theory, and, as an application to Yokota theory, present a refined formula of the complex volumes for any twist knots.

We prove an upper bound for the evaluation of all classical SU₂ spin networks conjectured by Garoufalidis and van der Veen. This implies one half of the analogue of the volume conjecture which they proposed for classical spin networks. We are also able to obtain the other half, namely, an exact determination of the spectral radius, for the special class of generalized drum graphs. Our proof uses a version of Feynman diagram calculus which we developed as a tool for the interpretation of the symbolic method of classical invariant theory, in a manner which is rigorous yet true to the spirit of the classical literature.

Yokota suggested an optimistic limit method of the Kashaev invariants of hyperbolic knots and showed it determines the complex volumes of the knots. His method is very effective and gives almost combinatorial method of calculating the complex volumes. However, to describe the triangulation of the knot complement, he restricted his method to knot diagrams with certain conditions. Although these restrictions are general enough for any hyperbolic knots, we have to select a good diagram of the knot to apply his theory. In this paper, we suggest more combinatorial way to calculate the complex volumes of hyperbolic links using the modified optimistic limit method. This new method works for any link diagrams, and it is more intuitive, easy to handle and has natural geometric meaning.

This note is dedicated to the memory of Slavik Jablan, and the last work we had together on the computation and properties of Cherns–Simons invariant of knots and links.

We define invariants for colored oriented spatial graphs by generalizing CM invariants [F. Costantino and J. Murakami, On $SL(2,C)$ quantum $6j$-symbols and their relation to the hyperbolic volume, Quantum Topol.4 (2013) 303–351], which were defined via non-integral highest weight representations of ${{𝒰}}_q(s{l}_2)$. We apply the same method used to define Yokota’s invariants, and we call these invariants Yokota type invariants. Then, we propose a volume conjecture of the Yokota type invariants of plane graphs, which relates to volumes of hyperbolic polyhedra corresponding to the graphs, and check it numerically for some square pyramids and pentagonal pyramids.

The Chen–Yang volume conjecture [Q. Chen and T. Yang, Volume conjectures for the Reshetikhin–Turaev and the Turaev–Viro invariants, preprint (2015), arXiv:1503.02547] states that the growth rate of the Turaev–Viro invariants of a compact oriented $3$-manifold determines its simplicial volume. In this paper, we prove that the Chen–Yang conjecture is stable under $(2n+1,2)$-cabling.

In this paper, in an attempt to extend an earlier work of Lück, we construct a knot invariant with parameter in C* by using the fundamental L²-representation of the fundamental group of the knot complement, which may be thought of as an L²-analogue of the usual Alexander polynomial of the knot in S³. When restricted to U(1) parameters, we interpret this invariant in terms of the U(1) twisted L²-Reidemeister torsion. We also show that this L²-invariant depends only on the norm |t| for t ∈ C*. In particular, this implies an unexpected rigidity property of the U(1) twisted L²-torsion on a knot complement. A possible relationship with the volume conjecture is discussed.

The volume conjecture and its generalizations say that the colored Jones polynomial corresponding to the N-dimensional irreducible representation of sl(2;ℂ) of a (hyperbolic) knot evaluated at exp(c/N) grows exponentially with respect to N if one fixes a complex number c near . On the other hand if the absolute value of c is small enough, it converges to the inverse of the Alexander polynomial evaluated at exp c. In this paper we study cases where it grows polynomially.

In this paper, by using the regulator map of Beilinson-Deligne on a curve, we show that the quantization condition posed by Gukov is true for the SL₂(ℂ) character variety of the hyperbolic knot in S³. Furthermore, we prove that the corresponding ℂ*-valued closed 1-form is a secondary characteristic class (Chern-Simons) arising from the vanishing first Chern class of the flat line bundle over the smooth part of the character variety, where the flat line bundle is the pullback of the universal Heisenberg line bundle over ℂ* × ℂ*. Based on this result, we give a reformulation of Gukov's generalized volume conjecture from a motivic perspective.

We examine the volume conjecture for the HOMFLY polynomial instead of the Jones polynomial. For the figure-eight knot, we explicitly obtain the limits of the colored HOMFLY polynomial.

No abstract received.