Hyperbolic Links Research Articles

Bipyramids and bounds on volumes of hyperbolic links

We present two methods to decompose a link complement into bipyramids, which have a variety of applications. Here, we apply them to obtain new upper bounds on volume for hyperbolic link complements in terms of the combinatorics of their projections.

Volume and determinant densities of hyperbolic rational links

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 [Formula: see text]. We construct sequences of alternating knots whose volume and determinant densities both converge to any [Formula: see text]. We also investigate the distributions of volume and determinant densities for hyperbolic rational links, and establish upper bounds and density results for these invariants.

The spectra of volume and determinant densities of links

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 voct, where voct is the volume of the ideal hyperbolic octahedron. We show that the set of volume densities is dense in [0,voct]. The determinant density of a link K is 2πlog⁡det⁡(K)/c(K). We prove that the closure of the set of determinant densities contains the set [0,voct].

Stable maps and branched shadows of 3-manifolds

In the early 1990s, Turaev introduced the notion of shadows as a combinatorial presentation of both 4 and 3-manifolds. Later, Costantino–Thurston revealed a strong relation between the Stein factorizations of stable maps of 3-manifolds into the real plane and the shadows of the manifolds. In fact, a shadow can be seen locally as the Stein factorization of a stable map. In this paper, we define the notion of stable map complexity for a compact orientable 3-manifold bounded by (possibly empty) tori counting, with some weights, the minimal number of singular fibers of codimension 2 of stable maps into the real plane, and prove that this number equals the minimal number of vertices of its branched shadows. In consequence, we give a complete characterization of hyperbolic links in the 3-sphere whose exteriors have stable map complexity 1 in terms of Dehn surgeries, and also give an observation concerning the coincidence of the stable map complexity and shadow complexity using estimations of hyperbolic volumes.

OPTIMISTIC LIMITS OF THE COLORED JONES POLYNOMIALS AND THE COMPLEX VOLUMES OF HYPERBOLIC 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.

Intercusp geodesics and the invariant trace field of hyperbolic 3-manifolds

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.

A geometrically bounding hyperbolic link complement

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.

Combinatorics of link diagrams and volume

We show that the volumes of certain hyperbolic A-adequate links can be bounded (above and) below in terms of two diagrammatic quantities: the twist number and the number of certain alternating tangles in an A-adequate diagram. We then restrict our attention to plat closures of certain braids, a rich family of links whose volumes can be bounded in terms of the twist number alone. Furthermore, in the absence of special tangles, our volume bounds can be expressed in terms of a single stable coefficient of the colored Jones polynomial. Consequently, we are able to provide a new collection of links that satisfy a Coarse Volume Conjecture.

Optimistic limits of Kashaev invariants and complex volumes of hyperbolic links

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.

Algebraic invariants, mutation, and commensurability of link complements

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.

Quasifuchsian state surfaces

This paper continues our study, initiated in [arXiv:1108.3370], of essential state surfaces in link complements that satisfy a mild diagrammatic hypothesis (homogeneously adequate). For hyperbolic links, we show that the geometric type of these surfaces in the Thurston trichotomy is completely determined by a simple graph--theoretic criterion in terms of a certain spine of the surfaces. For links with A- or B-adequate diagrams, the geometric type of the surface is also completely determined by a coefficient of the colored Jones polynomial of the link.

The closure of a random braid is a hyperbolic link

The closure of a random braid is a hyperbolic link

Quandle homology and complex volume

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.

The link volume of 3–manifolds

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.

The link volumes of some prism manifolds

In [13] the second named author and Yamashita defined the link volume, an invariant of closed orientable 3–manifolds that measures how efficiently a given manifold can be represented as a branched cover of S . We use the notation M d ! .S;L/ to denote a covering projection from M to S , branched along L and of degree d . We restrict to the case where L is a hyperbolic link. Then the complexity of M d ! .S;L/ is defined to be d Vol.S nL/, that is, the degree of the cover times the volume of the complement of the branch set. The link volume of a closed orientable 3–manifold M is denoted LinkVol.M / and defined to be the infimum of the complexities of all covers (of all possible degrees) M d ! .S;L/, that is,

HOMOLOGICALLY PERIPHERAL HYPERBOLIC LINKS WHICH ARE NOT PARTIALLY PERIPHERAL

Partially peripheral 3-manifolds and homologically peripheral 3-manifolds are defined by generalizing the notion of totally peripheral 3-manifolds. The aim of this paper is to provide a method for constructing hyperbolic links in 3-manifolds whose exteriors are not partially peripheral but homologically peripheral, and to prove that the connected sum with a homology 3-sphere enables every closed connected orientable 3-manifold to contain infinitely many such hyperbolic links.

The 27 Possible Intrinsic Symmetry Groups of Two-Component Links

We consider the “intrinsic” symmetry group of a two-component link L, defined to be the image ∑(L) of the natural homomorphism from the standard symmetry group MCG(S3, L) to the product MCG(S3) × MCG(L). This group, first defined by Whitten in 1969, records directly whether L is isotopic to a link L′ obtained from L by permuting components or reversing orientations; it is a subgroup of Γ2, the group of all such operations. For two-component links, we catalog the 27 possible intrinsic symmetry groups, which represent the subgroups of Γ2 up to conjugacy. We are able to provide prime, nonsplit examples for 21 of these groups; some are classically known, some are new. We catalog the frequency at which each group appears among all 77,036 of the hyperbolic two-component links of 14 or fewer crossings in Thistlethwaite’s table. We also provide some new information about symmetry groups of the 293 non-hyperbolic two-component links of 14 or fewer crossings in the table.

Planar open books with four binding components

We study an explicit construction of planar open books with four binding components on any three-manifold which is given by integral surgery on three component pure braid closures. This construction is general, indeed any planar open book with four binding components is given this way. Using this construction and results on exceptional surgeries on hyperbolic links, we show that any contact structure of S^3 supports a planar open book with four binding components, determining the minimal number of binding components needed for planar open books supporting these contact structures. In addition, we study a class of monodromies of a planar open book with four binding components in detail. We characterize all the symplectically fillable contact structures in this class and we determine when the Ozsvath-Szabo contact invariant vanishes. As an application, we give an example of a right-veering diffeomorphism on the four-holed sphere which is not destabilizable and yet supports an overtwisted contact structure. This provides a counterexample to a conjecture of Honda, Kazez, Matic from arXiv:0609734 .

SL2(ℂ)-character variety of a hyperbolic link and regulator

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.

On Hyperbolic 3‐Manifolds Obtained by Dehn Surgery on Links

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 Ld+1 of d + 1 components in Cavicchioli and Paoluzzi (2000).