Related papers: The $A$-Polynomial and Knot Volume
We show that for a large class of hyperbolic knots and links, we can determine bounds on the volume of the link complement from combinatorial information given by a link diagram. Specifically, there is a universal constant C such that if a…
We confirm the AJ conjecture [Ga04] that relates the A-polynomial and the colored Jones polynomial for those hyperbolic knots satisfying certain conditions. In particular, we show that the conjecture holds true for some classes of…
The work of J{\o}rgensen and Thurston shows that there is a finite number N(v) of orientable hyperbolic 3-manifolds with any given volume v. In this paper, we construct examples showing that the number of hyperbolic knot complements with a…
Given a knot in 3-space, one can associate a sequence of Laurrent polynomials, whose $n$th term is the $n$th colored Jones polynomial. The Generalized Volume Conjecture states that the value of the $n$-th colored Jones polynomial at $\exp(2…
We conjecture formulae of the colored superpolynomials for a class of twist knots $K_p$ where p denotes the number of full twists. The validity of the formulae is checked by applying differentials and taking special limits. Using the…
We propose a conjecture to compute the all-order asymptotic expansion of the colored Jones polynomial of the complement of a hyperbolic knot, J_N(q = exp(2u/N)) when N goes to infinity. Our conjecture claims that the asymptotic expansion of…
We prove that for hyperbolic fibered knots in any closed, connected, oriented 3-manifold the volume and genus are unrelated. As an application we answer a question of Hirose, Kalfagianni, and Kin about volumes of mapping tori that are…
Every oriented closed geodesic on the modular surface has a canonically associated knot in its unit tangent bundle coming from the periodic orbit of the geodesic flow. We study the volume of the associated knot complement with respect to…
We study a twisted Alexander polynomial naturally associated to a hyperbolic knot in an integer homology 3-sphere via a lift of the holonomy representation to SL(2, C). It is an unambiguous symmetric Laurent polynomial whose coefficients…
We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect non-trivially. Moreover it is shown that the intersection is (at least includes) the set of…
This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses that arise naturally in the study…
In hep-th/9805025, a result for the symmetric 3-loop massive tetrahedron in 3 dimensions was found, using the lattice algorithm PSLQ. Here we give a more general formula, involving 3 distinct masses. A proof is devised, though it cannot be…
We show that the proportion of hyperbolic knots among all of the prime knots of $n$ or fewer crossings does not converge to $1$ as $n$ approaches infinity. Moreover, we show that if $K$ is a nontrivial knot then the proportion of satellites…
We recently discovered a relationship between the volume density spectrum and the determinant density spectrum for infinite sequences of hyperbolic knots. Here, we extend this study to new quantum density spectra associated to quantum…
For a hyperbolic knot $K$ in $S^3$, the adjoint hyperbolic torsion polynomial $\mathcal T^{\mathrm{Ad}}_K(t) \in \mathbb C[t^{\pm 1}]$ is defined as a normalization of the twisted Alexander polynomial of $K$ associated with the…
This is an introduction to the Volume Conjecture and its generalizations for nonexperts. The Volume Conjecture states that a certain limit of the colored Jones polynomial of a knot would give the volume of its complement. If we deform the…
We study three-dimensional Chern-Simons theory with complex gauge group SL(2,C), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. We show that, in the presence of a single…
We show that there exist infinitely many examples of pairs of knots, K_1 and K_2, that have no epimorphism $\pi_1(S^3\setminus K_1) \to \pi_1(S^3\setminus K_2)$ preserving peripheral structure although their A-polynomials have the…
We study the knot invariant based on the quantum dilogarithm function. This invariant can be regarded as a non-compact analogue of Kashaev's invariant, or the colored Jones invariant, and is defined by an integral form. The 3-dimensional…
Let $D$ be a diagram of an alternating knot with unknotting number one. The branched double cover of $S^3$ branched over $D$ is an L-space obtained by half integral surgery on a knot $K_D$. We denote the set of all such knots $K_D$ by…