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In this paper, we characterize non-hyperbolic 3-component links in the 3-sphere whose exteriors contain essential 3-punctured spheres with non-integral boundary slopes. We also show the existence of embeddings of some multibranched surfaces in the 3-sphere which satisfy some homological conditions to be embedded in the 3-sphere., 11 pages, 7 figures

In a group, a generalized torsion element is a non-identity element whose some non-empty finite product of its conjugates yields the identity. Such an element is an obstruction for a group to be bi-orderable. We show that the Weeks manifold, the figure-eight sister manifold, and the complement of Whitehead sister link admit generalized torsion elements in their fundamental groups. In particular, the Whitehead sister link, which is the pretzel link of type (-2,3,8), can be generalized to hyperbolic pretzel links of type (-2,3,2n), n>3. These give the first examples of hyperbolic links whose link groups admit generalized torsion elements., 8 pages, 7 figures

Changes to gluing patterns for fully augmented links are shown to result in generalized fully augmented links. The first changes considered result in [math] –tangle surgeries on hyperbolic fully augmented links that produce hyperbolic generalized fully augmented links. These surgeries motivate the definition of nested links, and a characterization of hyperbolic nested links is given. Finally, the geometry of nested links is compared to that of fully augmented links.

The optimistic limit is a mathematical formulation of the classical limit, which is a physical method to estimate the actual limit by using the saddle-point method of a certain potential function. The original optimistic limit of the Kashaev invariant was formulated by Yokota, and a modified formulation was suggested by the author and others. This modified version is easier to handle and more combinatorial than the original one. On the other hand, it is known that the Kashaev invariant coincides with the evaluation of the colored Jones polynomial at a certain root of unity. This optimistic limit of the colored Jones polynomial was also formulated by the author and others, but it is very complicated and needs many unnatural assumptions. In this article, we suggest a modified optimistic limit of the colored Jones polynomial, following the idea of the modified optimistic limit of the Kashaev invariant, and show that it determines the complex volume of a hyperbolic link. Furthermore, we show that this optimistic limit coincides with the optimistic limit of the Kashaev invariant modulo $4{\it\pi}^{2}$. This new version is easier to handle and more combinatorial than the old version, and has many advantages over the modified optimistic limit of the Kashaev invariant. Because of these advantages, several applications have already appeared and more are in preparation.

Given a cusped hyperbolic 3-manifold with finite volume, we define two types of complex parameters which capture geometric information about the preimages of geodesic arcs traveling between cusp cross-sections. We prove that these parameters are elements of the invariant trace field of the manifold, providing a connection between the intrinsic geometry of a 3-manifold and its number-theoretic invariants. Further, we explore the question of choosing a minimal collection of arcs and associated parameters to generate the field. We prove that for a tunnel number $k$ manifold it is enough to choose $3k$ specific parameters. For many hyperbolic link complements, this approach allows one to compute the field from a link diagram. We also give examples of infinite families of links where a single parameter can be chosen to generate the field, and the polynomial for it can be constructed from the link diagram as well.

We introduce a new homology theory of quandles, called simplicial quandle homology, which is quite different from quandle homology developed by Carter et al. We construct a homomorphism from a quandle homology group to a simplicial quandle homology group. As an application, we obtain a method for computing the complex volume of a hyperbolic link only from its diagram., Comment: 27 pages, 11 figures; added Lemma 7.2

We view closed orientable 3-manifolds as covers of S^3 branched over hyperbolic links. For a p-fold cover M \to S^3, branched over a hyperbolic link L, we assign the complexity p Vol(S^3 minus L) (where Vol is the hyperbolic volume). We define an invariant of 3-manifolds, called the link volume and denoted LV, that assigns to a 3-manifold M the infimum of the complexities of all possible covers M \to S^3, where the only constraint is that the branch set is a hyperbolic link. Thus the link volume measures how efficiently M can be represented as a cover of S^3. We study the basic properties of the link volume and related invariants, in particular observing that for any hyperbolic manifold M, Vol(M) < LV(M). We prove a structure theorem that is similar to (and relies on) the celebrated theorem of Jorgensen and Thurston. This leads us to conjecture that, generically, the link volume of a hyperbolic 3-manifold is much bigger than its volume. Finally we prove that the link volumes of the manifolds obtained by Dehn filling a manifold with boundary tori are linearly bounded above in terms of the length of the continued fraction expansion of the filling curves., 27 pages, several color figures (viewable in black and white)

We provide an algorithm to determine the Heegaard genus of simple 3-manifolds with non-empty boundary. More generally, we supply an algorithm to determine (up to ambient isotopy) all the Heegaard splittings of any given genus for the manifold. As a consequence, the tunnel number of a hyperbolic link is algorithmically computable. Our techniques rely on Rubinstein's work on almost normal surfaces, and also a new structure called a partially flat angled ideal triangulation., 23 pages, 14 figures

We prove that any Bonahon-Siebenmann family of Conway spheres for a hyperbolic link is associated to an ideal point of the character variety of the link., Comment: 19 pages, 2 figures

In this paper, we study the SL2(C) character variety of a hyperbolic link in S 3 . We analyze a special smooth projective variety Y h arising from some 1-dimensional irreducible slices on the character variety. We prove that a natural symbol obtained from these 1-dimensional slices is a torsion in K2(C(Y h )). By using the regulator map from K2 to the corresponding Deligne cohomology, we get some variation formulae on some Zariski open subset of Y h . From this we give some discussions on a possible parametrized volume conjecture for both hyperbolic links and knots.

It was previously shown by the second author that every knot in $S^3$ is ambient isotopic to one component of a two-component, alternating, hyperbolic link. In this paper, we define the alternating volume of a knot $K$ to be the minimum volume of any link $L$ in a natural class of alternating, hyperbolic links such that $K$ is ambient isotopic to a component of $L$. Our main result shows that the alternating volume of a knot is coarsely equivalent to the twist number of a knot., 15 pages, 14 figures

A closed 3-manifold M is said to be hyperelliptic if it has an involution τ such that the quotient space of M by the action of τ is homeomorphic to the standard 3-sphere. We show that the hyperbolic football manifolds of Emil Molnar [12] are hyperelliptic. Then we determine the isometry groups of such manifolds. Another consequence is that the unique hyperbolic dodecahedral and icosahedral 3-space forms with first homology group ℤ35 (constructed by I. Prok in [16], on the basis of a principal algorithm due to Emil Molnar [13], and by Richardson and Rubinstein in [18]) are also hyperelliptic.

We prove that every finite group is the orientation-preserving isometry group of the complement of a hyperbolic link in the 3-sphere., Only the footnote for the funding institutions have been modified

We show that one can interweave an unknot into any non-alternating connected projection of a link so that the resulting augmented projection is alternating., Comment: 5 pages, 7 figures

In this paper we apply the twisted Alexander polynomial to study the fibering and genus detecting problems for oriented links. In particular we generalize a conjecture of Dunfield, Friedl and Jackson on the torsion polynomial of hyperbolic knots to hyperbolic links, and confirm it for an infinite family of hyperbolic 2-bridge links. Moreover we consider a similar problem for parabolic representations of 2-bridge link groups., Comment: 19 pages, 3 figures. Title changed. Section 3 (Finiteness theorems for alternating links) of the old version is removed, since there is a gap in the proof of Theorem 3.3. A new section on parabolic representations (Section 5) is added in the new version

We utilize ideal bipyramids to obtain new upper bounds on volume for hyperbolic link complements in terms of the combinatorics of their projections., 17 pages, 10 figures

We define A-polynomial n-tuple for a link of n-components and apply them to construct hyperbolic link manifolds with non-integral traces.

This paper introduces a new approach to finding knots and links with hidden symmetries using "hidden extensions", a class of hidden symmetries defined here. We exhibit a family of tangle complements in the ball whose boundaries have symmetries with hidden extensions, then we further extend these to hidden symmetries of some hyperbolic link complements., 16 pages, 4 figures. Many changes suggested by a referee, including overhauled exposition in Section 1

The volume density of a hyperbolic link K is defined to be the ratio of the hyperbolic volume of K to the crossing number of K . We show that there are sequences of non-alternating links with volume density approaching v oct , where v oct is the volume of the ideal hyperbolic octahedron. We show that the set of volume densities is dense in [ 0 , v oct ] . The determinant density of a link K is 2 π log ⁡ det ⁡ ( K ) / c ( K ) . We prove that the closure of the set of determinant densities contains the set [ 0 , v oct ] .

Let M be a hyperbolic three-manifold which is an n-fold cyclic branched covering of a hyperbolic link L in the three-sphere, or more precisely, of the hyperbolic three-orbifold \(\mathcal{O}_n (L)\) whose underlying topological space is the three-sphere and whose singular set, of branching index n, the link L. We say that M has no hidden symmetries (with respect to the given branched covering) if the isometry group of M is the lift of (a subgroup of) the isometry group of the hyperbolic orbifold \(\mathcal{O}_n (L)\) (which is isomorphic to the symmetry group of the link L). It follows from Thurston's hyperbolic surgery theorem that M has no hidden symmetries if n is sufficiently large. Our main result is an explicit numerical version of this fact: we give a constant, in terms of the volume of the complement of L, such that M has no hidden symmetries for all n larger than this constant; we show by examples that a universal constant working for all hyperbolic knots or links does not exist. We give also some results on the possible orders and the structure of the isometry group of M. Finally, we construct sets of four different π-hyperbolic knots which have the same two-fold branched covering (a hyperbolic three-manifold); it is an interesting question for how many different π-hyperbolic knots (or links) this may happen (in the case of hyperbolic knots, for arbitrarily many).

A finite-volume hyperbolic 3-manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here an example of non-compact, finite-volume hyperbolic 3-manifold that geometrically bounds. The 3-manifold is the complement of a link with eight components, and its volume is roughly equal to 29.311., 23 pages, 19 figures

There exist in the literature many examples of different knots or links with homeomorphic cyclic branched coverings. In this paper we study the case of hyperbolic links. We show that a hyperbolic link with r components, if r does not divide n , is determined by its n -fold cyclic branched coverings.

It is well known that different knots or links in the 3-sphere can have homeomorphic n-fold cyclic branched coverings. We consider the following problem: for which values of nis a knot of link determined by itsn-fold cyclic branched covering? We consider the class of hyperbolic resp.2π/n-hyperbolic links. The isometry or symmetry groups of such links are finite, and their n-fold branched coverings are hyperbolic 3-manifolds. Our main result states that if ndoes not divide the order of the finite symmetry group of such a link, then the link is determined by its n-fold branched covering. In a sense, the result is best possible; the key argument of its proof is algebraic using some basic result about finite p-groups. The main result applies, for example, to the cyclic branched coverings of the 2-bridge links; in particular, it gives a classification of the maximally symmetricD6-manifolds which are exactly the 3-fold branched coverings of the 2-bridge links.

The invariant of a link in three-sphere, associated with the cyclic quantum dilogarithm, depends on a natural number N. By the analysis of particularexamples, it is argued that, for a hyperbolic knot (link), the absolute valueof this invariant grows exponentially at large N, the hyperbolic volume of the knot (link) complement being the growth rate.

We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds for surfaces with meridional boundary: such a surface either has large negative Euler characteristic, or is an n-punctured sphere visible in the diagram., 17 pages, 11 figures

We give a complete classification of the unknotting tunnels in 2-bridge link complements, proving that only the upper and lower tunnels are unknotting tunnels. Moreover, we show that the only strongly parabolic tunnels in 2-cusped hyperbolic 3-manifolds are exactly the upper and lower tunnels in 2-bridge knot and link complements. From this, it follows that the upper and lower tunnels in 2-bridge knot and link complements must be isotopic to geodesics of length at most ln(4), where length is measured relative to maximal cusps. Moreover, the four dual unknotting tunnels in a 2-bridge knot complement, which together with the upper and lower tunnels form the set of all known unknotting tunnels for these knots, must each be homotopic to a geodesic of length at most 6ln(2).

The volume density of a hyperbolic link is defined as the ratio of hyperbolic volume to crossing number. We study its properties and a closely-related invariant called the determinant density. It is known that the sets of volume densities and determinant densities of links are dense in the interval [0,v_{oct}]. We construct sequences of alternating knots whose volume and determinant densities both converge to any x in [0,v_{oct}]. We also investigate the distributions of volume and determinant densities for hyperbolic rational links, and establish upper bounds and density results for these invariants., Comment: 8 pages, 4 figures

Adams and Reid produced an upper bound for the length of a shortest closed geodesic in a hyperbolic knot or link complement in closed 3- manifolds which do not admit any Riemannian metric of negative curvature. We demonstrate that the length of an n th shortest closed geodesic in such manifolds is also bounded above for n > 1 and produce explicit upper bound on this length.

Two of the main methods for the construction of closed orientable 3-manifolds, and in particular of hyperbolic 3-manifolds, are surgery on links and branched coverings of links. If the link is hyperbolic, i.e., has hyperbolic complement, by results of Thurston most of the resulting 3-manifolds are hyperbolic. In the present paper, for a fixed integer n ⩾ 2, we consider hyperbolic 3-manifolds M n , k which are cyclic n -fold branched coverings of a hyperbolic link with two components. Our main theorem relates the classification up to isometry or homeomorphism of these manifolds to the symmetry group of the link and allows a complete classification of these manifolds in various cases; as an example, we consider cyclic branched coverings of the Whitehead link. The classification resembles the classification of lens spaces which are the cyclic branched coverings of the Hopf link (which is not hyperbolic, however); it generalizes to links with more than two components. For the proof, which consists in a mixture of geometric and algebraic arguments, we extend methods used in [5] in a special case.

We construct a family of hyperbolic link complements by gluing tangles along totally geodesic four-punctured spheres, then investigate the commensurability relation among its members. Those with different volume are incommensurable, distinguished by their scissors congruence classes. Mutation produces arbitrarily large finite subfamilies of nonisometric manifolds with the same volume and scissors congruence class. Depending on the choice of mutation, these manifolds may be commensurable or incommensurable, distinguished in the latter case by cusp parameters. All have trace field Q(i,\sqrt{2}), but some have integral traces while others do not., Minor changes following referee's suggestions

In arxiv:1205.1274 Rieck and Yamashita defined the link volume of 3-manifolds and studied some of its basic properties. Many of these properties are similar to the corresponding properties of the hyperbolic volume. In this paper we calculate the link volume of an infinite family of prism manifolds. As a corollary, we show that (in contrast to the hyperbolic volume) the link volume is not finite-to-one., Comment: 15 pages, final version. To appear in Algebraic and Geometric Topology (numbering and pagination may differ from published version)

We study the algebraic and geometric structures for closed orientable 3 -manifolds obtained by Dehn surgery along the family of hyperbolic links with certain surgery coefficients and moreover, the geometric presentations of the fundamental group of these manifolds. We prove that our surgery manifolds are 2 -fold cyclic covering of 3 -sphere branched over certain link by applying the Montesinos theorem in Montesinos-Amilibia (1975). In particular, our result includes the topological classification of the closed 3 -manifolds obtained by Dehn surgery on the Whitehead link, according to Mednykh and Vesnin (1998), and the hyperbolic link 𝐿 𝑑 + 1 of 𝑑 + 1 components in Cavicchioli and Paoluzzi (2000).

R.M. Kashaev conjectured that the asymptotic behavior of his link invariant, which equals the colored Jones polynomial evaluated at a root of unity, determines the hyperbolic volume of any hyperbolic link complement. We observe numerically that for knots $6_3$, $8_9$ and $8_{20}$ and for the Whitehead link, the colored Jones polynomials are related to the hyperbolic volumes and the Chern-Simons invariants and propose a complexification of Kashaev's conjecture., 14 pages, 9 figures. Added some calculations

We study the topological structure and the homeomorphism problem for closed 3-manifolds M(n,k) obtained by pairwise identifications of faces in the boundary of certain polyhedral 3-balls. We prove that they are (n/d)-fold cyclic coverings of the 3-sphere branched over certain hyperbolic links of d+1 components, where d= (n/k). Then we study the closed 3-manifolds obtained by Dehn surgeries on the components of these links.

If a hyperbolic link has a prime alternating diagram D, then we show that the link complement's volume can be estimated directly from D. We define a very elementary invariant of the diagram D, its twist number t(D), and show that the volume lies between v_3(t(D) - 2)/2 and v_3(16t(D) - 16), where v_3 is the volume of a regular hyperbolic ideal 3-simplex. As a consequence, the set of all hyperbolic alternating and augmented alternating link complements is a closed subset of the space of all complete finite volume hyperbolic 3-manifolds, in the geometric topology. The appendix by Ian Agol and Dylan Thurston, which was written after the first version of this paper was distributed, improves the upper bound on volume to v_3(10t(D) - 10). In addition, examples of alternating links are given which asymptotically achieve this bound., Comment: 30 pages, 17 figures; contains an appendix by Ian Agol and Dylan Thurston

Let $\Sigma$ be a hyperbolic link with $m$ components in a 3-dimensional manifold $X$. In this paper, we will show that the moduli space of marked hyperbolic cone structures on the pair $(X, \Sigma)$ with all cone angle less than $2\pi /3$ is an $m$-dimensional open cube, parameterized naturally by the $m$ cone angles. As a corollary, we will give a proof of a special case of Thurston's geometrization theorem for orbifolds., Comment: 29 pages