Hyperbolic Knots Research Articles

Cabling in the Skyrme–Faddeev model

The Skyrme–Faddeev model is a three-dimensional nonlinear field theory that has topological soliton solutions, called hopfions, which are novel string-like solutions taking the form of knots and links. Solutions found thus far take the form of torus knots and links of these, however torus knots form only a small family of known knots. It is an open question whether any non-torus knot hopfions exist. In this paper we present a construction of knotted fields with the form of cable knots to which an energy minimization scheme can be applied. We find the first known hopfions which do not have the form of torus knots, but instead take the form of cable and hyperbolic knots.

Recognizing a relatively hyperbolic group by its Dehn fillings

Dehn fillings for relatively hyperbolic groups generalize the topological Dehn surgery on a non-compact hyperbolic $3$-manifold such as a hyperbolic knot complement. We prove a rigidity result saying that if two non-elementary relatively hyperbolic groups without suitable splittings have sufficiently many isomorphic Dehn fillings, then these groups are in fact isomorphic. Our main application is a solution to the isomorphism problem in the class of non-elementary relatively hyperbolic groups with residually finite parabolic groups and with no suitable splittings.

Partial classification of Lorenz knots: Syllable permutations of torus knots words

We define families of aperiodic words associated to Lorenz knots that arise naturally as syllable permutations of symbolic words corresponding to torus knots. An algorithm to construct symbolic words of satellite Lorenz knots is defined. We prove, subject to the validity of a previous conjecture, that Lorenz knots coded by some of these families of words are hyperbolic, by showing that they are neither satellites nor torus knots and making use of Thurston’s theorem. Infinite families of hyperbolic Lorenz knots are generated in this way, to our knowledge, for the first time. The techniques used can be generalized to study other families of Lorenz knots.

On the Kashaev invariant and the twisted Reidemeister torsion of two-bridge knots

It is conjectured that, in the asymptotic expansion of the Kashaev invariant of a hyperbolic knot, the first coefficient is represented by the complex volume of the knot complement, and the second coefficient is represented by a constant multiple of the square root of the twisted Reidemeister torsion associated with the holonomy representation of the hyperbolic structure of the knot complement. In particular, this conjecture has been rigorously proved for some simple hyperbolic knots, for which the second coefficient is presented by a modification of the square root of the Hessian of the potential function of the hyperbolic structure of the knot complement. In this paper, we define an invariant of a parametrized knot diagram as a modification of the Hessian of the potential function obtained from the parametrized knot diagram. Further, we show that this invariant is equal (up to sign) to a constant multiple of the twisted Reidemeister torsion for any two-bridge knot. 57M27;

All order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials

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 = e^{\frac{2u}{N}}) when N \rightarrow \infty . Our conjecture claims that the asymptotic expansion of the colored Jones polynomial is a formal wave function of an integrable system whose semiclassical spectral curve \mathcal{C} would be the \mathrm{SL}_2(\mathbb{C}) character variety of the knot (the A-polynomial), and is formulated in the framework of the topological recursion. It takes as starting point the proposal made recently by Dijkgraaf, Fuji and Manabe (who kept only the perturbative part of the wave function, and found some discrepancies), but it also contains the non-perturbative parts, and solves the discrepancy problem. These non-perturbative corrections are derivatives of Theta functions associated to \mathcal{C} . For a large class of knots, this expansion is still in powers of 1/N due to the special properties of A-polynomials. We provide a detailed check of our proposal for the figure-eight knot and the once-punctured torus bundle L^2R . We also present a heuristic argument inspired from the case of torus knots, for which knot invariants can be computed by a matrix model.

A Note on Colored HOMFLY Polynomials for Hyperbolic Knots from WZW Models

Using the correspondence between Chern-Simons theories and Wess-Zumino-Witten models we present the necessary tools to calculate colored HOMFLY polynomials for hyperbolic knots. For two-bridge hyperbolic knots we derive the colored HOMFLY invariants in terms of crossing matrices of the underlying Wess-Zumino-Witten model. Our analysis extends previous works by incorporating non-trivial multiplicities for the primaries appearing in the crossing matrices, so as to describe colorings of HOMFLY invariants beyond the totally symmetric or anti-symmetric representations of SU(N). The crossing matrices directly relate to 6j-symbols of the quantum group U_q(su(N)). We present powerful methods to calculate such quantum 6j-symbols for general N. This allows us to determine previously unknown colored HOMFLY polynomials for two-bridge hyperbolic knots. We give explicitly the HOMFLY polynomials colored by the representation {2,1} for two-bridge hyperbolic knots with up to eight crossings. Yet, the scope of application of our techniques goes beyond knot theory; e.g., our findings can be used to study correlators in Wess-Zumino-Witten conformal field theories or -- in the limit to classical groups -- to determine color factors for Yang Mills amplitudes.

How to include fermions into general relativity by exotic smoothness

This paper is two-fold. At first we will discuss the generation of source terms in the Einstein-Hilbert action by using (topologically complicated) compact 3-manifolds. There is a large class of compact 3-manifolds with boundary: a torus given as the complement of a (thickened) knot admitting a hyperbolic geometry, denoted as hyperbolic knot complements in the following. We will discuss the fermionic properties of this class of 3-manifolds, i.e. we are able to identify a fermion with a hyperbolic knot complement. Secondly we will construct a large class of space-times, the exotic $\mathbb{R}^{4}$, containing this class of 3-manifolds naturally. We begin with a topological trivial space, the $\mathbb{R}^{4}$, and change only the differential structure to obtain many nontrivial 3-manifolds. It is known for a long time that exotic $\mathbb{R}^{4}$'s generate extra sources of gravity (Brans conjecture) but here we will analyze the structure of these source terms more carefully. Finally we will state that adding a hyperbolic knot complement will result in the appearance of a fermion as source term in the Einstein-Hilbert action.

KNOTS ADMITTING SEIFERT-FIBERED SURGERIES OVER S2WITH FOUR EXCEPTIONAL FIBERS

In this paper, we construct infinite families of knots in <TEX>$S^3$</TEX> which admit Dehn surgery producing a Seifert-fibered space over <TEX>$S^2$</TEX> with four exceptional fibers. Also we show that these knots are turned out to be satellite knots, which supports the conjecture that no hyperbolic knot in <TEX>$S^3$</TEX> admits a Seifert-fibered space over <TEX>$S^2$</TEX> with four exceptional fibers as Dehn surgery.

Factorial growth rates for the number of hyperbolic 3-manifolds of a given volume

The work of Jørgensen and Thurston shows that there is a finite number N ( v ) N(v) of orientable hyperbolic 3 3 -manifolds with any given volume v v . In this paper, we construct examples showing that the number of hyperbolic knot complements with a given volume v v can grow at least factorially fast with v v . A similar statement holds for closed hyperbolic 3 3 -manifolds, obtained via Dehn surgery. Furthermore, we give explicit estimates for lower bounds of N ( v ) N(v) in terms of v v for these examples. These results improve upon the work of Hodgson and Masai, which describes examples that grow exponentially fast with v v . Our constructions rely on performing volume preserving mutations along Conway spheres and on the classification of Montesinos knots.

A kernel of a braid group representation yields a knot with trivial knot polynomials

We show that a non-trivial, non-central normal subgroup of the braid groups contains a braid whose closure is a hyperbolic knot with arbitrary large genus. This shows that non-faithfulness of a quantum representation implies that the corresponding quantum invariant fails to detect the unknot. The proof utilizes the Dehornoy ordering of the braid groups.

Half-integral finite surgeries on knots in S 3

Supposing that a hyperbolic knot in S 3 admits a finite surgery, Boyer and Zhang proved that the surgery slope must be either integral or half-integral, and they conjectured that the latter case does not happen. Using the correction terms in Heegaard Floer homology, we prove that if a hyperbolic knot in S 3 admits a half-integral finite surgery, then the knot must have the same knot Floer homology as one of the eight non-hyperbolic knots which are known to admit such surgeries, and the resulting manifold must be one of ten spherical space forms. As knot Floer homology carries a lot of information about the knot, this gives a strong evidence to Boyer–Zhang’s conjecture.

Knot complements, hidden symmetries and reflection orbifolds

In this article we examine the conjecture of Neumann and Reid that the only hyperbolic knots in the 3-sphere which admit hidden symmetries are the figure-eight knot and the two dodecahedral knots. Knots whose complements cover hyperbolic reflection orbifolds admit hidden symmetries, and we verify the Neumann-Reid conjecture for knots which cover small hyperbolic reflection orbifolds. We also show that a reflection orbifold covered by the complement of an AP knot is necessarily small. Thus when K is an AP knot, the complement of K covers a reflection orbifold exactly when K is either the figure-eight knot or one of the dodecahedral knots.

Hyperbolic L-space knots and exceptional Dehn surgeries

A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space. Like torus knots and Berge knots, many known L-space knots admit a Seifert fibered L-space surgery. We give a concrete example of a hyperbolic L-space knot which has no exceptional surgeries, in particular, no Seifert fibered surgeries.

Bridge number, Heegaard genus and non-integral Dehn surgery

We show there exists a linear function w: N->N with the following property. Let K be a hyperbolic knot in a hyperbolic 3-manifold M admitting a non-longitudinal S^3 surgery. If K is put into thin position with respect to a strongly irreducible, genus g Heegaard splitting of M then K intersects a thick level at most 2w(g) times. Typically, this shows that the bridge number of K with respect to this Heegaard splitting is at most w(g), and the tunnel number of K is at most w(g) + g-1.

Trigonometric identities and volumes of the hyperbolic twist knot cone-manifolds

We calculate the volumes of the hyperbolic twist knot cone-manifolds using the Schläfli formula. Even though general ideas for calculating the volumes of cone-manifolds are around, since there is no concrete calculation written, we present here the concrete calculations. We express the length of the singular locus in terms of the distance between the two axes fixed by two generators. In this way the calculation becomes easier than using the singular locus directly. The volumes of the hyperbolic twist knot cone-manifolds simpler than Stevedore's knot are known. As an application, we give the volumes of the cyclic coverings over the hyperbolic twist knots.

The 500 simplest hyperbolic knots

We identify all hyperbolic knots whose complements are in the census of orientable one-cusped hyperbolic manifolds with eight ideal tetrahedra. We also compute their Jones polynomials.

Networking Seifert surgeries on knots, III

How do Seifert surgeries on hyperbolic knots arise from those on torus knots? We approach this question from a networking viewpoint. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; two vertices are connected by an edge if one Seifert surgery is obtained from the other by a single twist along a trivial knot called a seiferter or along an annulus cobounded by seiferters. Successive twists along a "hyperbolic seiferter" or a "hyperbolic annular pair" produce infinitely many Seifert surgeries on hyperbolic knots. In this paper, we investigate Seifert surgeries on torus knots which have hyperbolic seiferters or hyperbolic annular pairs, and obtain results suggesting that such surgeries are restricted.

Neighbors of Seifert surgeries on a trefoil knot in the Seifert Surgery Network

A Seifert surgery is a pair $$(K, m)$$ of a knot $$K$$ in $$S^3$$ and an integer $$m$$ such that $$m$$ -Dehn surgery on $$K$$ results in a Seifert fiber space allowed to contain fibers of index zero. Twisting $$K$$ along a trivial knot called a seiferter for $$(K, m)$$ yields Seifert surgeries. We study Seifert surgeries obtained from those on a trefoil knot by twisting along their seiferters. Although Seifert surgeries on a trefoil knot are the most basic ones, this family is rich in variety. For any $$m \ne -2$$ it contains a successive triple of Seifert surgeries $$(K, m)$$ , $$(K, m +1)$$ , $$(K, m +2)$$ on a hyperbolic knot $$K$$ , e.g. $$17$$ -, $$18$$ -, $$19$$ -surgeries on the $$(-2, 3, 7)$$ pretzel knot. It contains infinitely many Seifert surgeries on strongly invertible hyperbolic knots none of which arises from the primitive/Seifert-fibered construction, e.g. $$(-1)$$ -surgery on the $$(3, -3, -3)$$ pretzel knot.

A cabling conjecture for knots in lens spaces

Closed $$3$$ -string braids admit many bandings to two-bridge links. By way of the Montesinos Trick, this allows us to construct infinite families of knots in the connected sum of lens spaces $$L(r,1) \# L(s,1)$$ that admit a surgery to a lens space for all pairs of integers $$(r,s)$$ except $$(0,0)$$ . These knots are typically hyperbolic. We also demonstrate that the previously known two families of examples of hyperbolic knots in non-prime manifolds with lens space surgeries of Eudave-Munoz–Wu and Kang all fit this construction. As such, we propose a generalization of the cabling conjecture of Gonzalez-Acuna–Short for knots in lens spaces.

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.