Figure-eight Knot Research Articles

Geometric conemanifold structures on 𝕋p/q, the result of p/q surgery in the left-handed trefoil knot 𝕋

As an example of the transitions between some of the eight geometries of Thurston, investigated in [M. T. Lozano and J. M. Montesinos-Amilibia, On the degeneration of some 3-manifold geometries via unit groups of quaternion algebras RACSAM 109 (2015) 669–715], we study the geometries supported by the conemanifolds obtained by surgery on the trefoil knot with singular set the core of the surgery. The geometric structures are explicitly constructed. The most interesting phenomenon is the transition from SL(2, ℝ)-geometry to S3-geometry through Nil-geometry. A plot of the different geometries is given, in the spirit of the analogous plot of Thurston for the geometries supported by surgeries on the figure-eight knot.

2-Adjacency between knots

We study the properties of a knot K to be 2-adjacent to another knot W by analyzing their Conway polynomials, Jones polynomials and Homfly polynomials and give some very useful conditions. We discuss whether each pair of knots can be 2-adjacent to each other, i.e. whether 2-adjacency is a symmetric relation. We discuss also whether the trivial knot, the trefoil knot and the figure-eight knot can be 2-adjacent to any knot in Rolfsen's table and the opposite cases, except for 934 it is not decided whether it is 2-adjacent to 41. Finally, we give some examples to answer I. Torisu's problem partly and etc.

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.

Complex hyperbolic geometry of the figure-eight knot

We show that the figure eight knot complement admits a uniformizable spherical CR structure, i.e. it occurs as the manifold at infinity of a complex hyperbolic orbifold. The uniformization is unique provided we require the peripheral subgroups to have unipotent holonomy.

Finite volume properly convex deformations of the figure-eight knot

In this paper we show that some open set of the representations of the fundamental group of figure-eight knot complement found in \cite{Ballas12a} are the holonomies of a family of finite volume properly convex projective structures on the figure-eight knot complement.

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.

Matrix integral expansion of colored Jones polynomials for figure-eight knot

In this note we examine a possible extension of the matrix integral representation of knot invariants beyond the class of torus knots. In particular, we study a representation of the SU(2) quantum Racah coefficients by double matrix integrals. We find that the Racah coefficients are mapped to expansion coefficients in some basis of double integrals. The transformed coefficients have a number of interesting algebraic properties.

A general method for deriving vector potentials produced by knotted solenoids

A general method for deriving exact expressions for vector potentials produced by arbitrarily knotted solenoids is presented. It consists of using simple physics ideas from magnetostatics to evaluate the magnetic field in a surrogate problem. The latter is obtained by modeling the knot with wire segments carrying steady currents on a cubical lattice. The expressions for a 31 (trefoil) and a 41 (figure-eight) knot are explicitly worked out. The results are of some importance in the study of the Aharonov–Bohm effect generalized to a situation in which charged particles moving through force-free regions are scattered by fluxes confined to the interior of knotted impenetrable tubes.

Inflations of ideal triangulations

Starting with an ideal triangulation of M∘, the interior of a compact 3-manifold M with boundary, no component of which is a 2-sphere, we provide a construction, called an inflation of the ideal triangulation, to obtain a strongly related triangulation of M itself. Besides a step-by-step algorithm for such a construction, we provide examples of an inflation of the two-tetrahedra ideal triangulation of the complement of the figure-eight knot in S3, giving a minimal triangulation, having ten tetrahedra, of the figure-eight knot exterior. As another example, we provide an inflation of the one-tetrahedron Gieseking manifold giving a minimal triangulation, having seven tetrahedra, of a non-orientable compact 3-manifold with Klein bottle boundary. Several applications of inflations are discussed.

Branched sperical CR structures on the complement of the figure-eight knot

Abstract. We obtain a branched spherical CR structure on the com- plement of the figure-eight knot whose holonomy representation was given in [4]. There are essentially two boundary unipotent representations from the complement of the figure-eight knot into PU(2,1), which we call ρ1 and ρ2. We make explicit some fundamental differences between these two representations. For instance, seeing the figure-eight knot complement as a surface bundle over the circle, the behavior of the fundamental group of the fiber under the representa- tion is a key difference between ρ1 and ρ2.

On Jørgensen numbers and their analogs for groups of figure-eight orbifolds

The Jorgensen, Gehring-Martin-Tan, and Tan numbers are defined for every two-generated subgroup of the group PSL(2,C). These numbers arise in necessary discreteness conditions for two-generated subgroups. The Jorgensen number equals 1 for the figure-eight knot group. We calculate the above numbers or give some two-sided bounds of them for this group and groups of hyperbolic orbifolds with singularities along the figure-eight knot.

Seifert volumes and dilogarithm identities

In this paper, we obtain new dilogarithm identities by computing the Seifert volume of manifolds obtained by Dehn surgery on the figure-eight knot in two different ways.

Links with small lattice stick numbers

Knots and links have been considered to be useful models for structural analysis of molecular chains such as DNA and proteins. One quantity that we are interested in for molecular links is the minimum number of monomers necessary for realizing them. In this paper we consider every link in the cubic lattice. The lattice stick number sL(L) of a link L is defined to be the minimum number of sticks required to construct a polygonal representation of the link in the cubic lattice. Huh and Oh found all knots whose lattice stick numbers are at most 14. They proved that only the trefoil knot 31 and the figure-eight knot 41 have lattice stick numbers of 12 and 14, respectively. In this paper we find all links with more than one component whose lattice stick numbers are at most 14. Indeed we prove combinatorically that , , , and any other non-split links have stick numbers of at least 15.

Knot polynomials in the first non-symmetric representation

We describe the explicit form and the hidden structure of the answer for the HOMFLY polynomial for the figure-8 and some other 3-strand knots in representation [21]. This is the first result for non-torus knots beyond (anti)symmetric representations, and its evaluation is far more complicated. We provide a whole variety of different arguments, allowing one to guess the answer for the figure-8 knot, which can be also partly used in more complicated situations. Finally we report the result of exact calculation for figure-8 and some other 3-strand knots based on the previously developed sophisticated technique of multi-strand calculations. We also discuss a formula for the superpolynomial in representation [21] for the figure-8 knot, which heavily relies on the conjectural form of superpolynomial expansion nearby the special polynomial point. Generalizations and details will be presented elsewhere.

TOTALLY GEODESIC SURFACES AND QUADRATIC FORMS

Let M be a compact, connected, irreducible, orientable 3-manifold with torus boundary. A closed, orientable, immersed, incompressible surface F in M with no incompressible annulus joining F and ∂M compresses in at most finitely many Dehn fillings M(α). It is known that there is no universal upper bound on the number of such fillings, independent of the surface, and the figure-eight knot complement is the first example of a manifold where this phenomenon occurs. In this paper, we show that the same behavior of the figure-eight knot complement is shared by other two cusped manifolds.

ON 2-ADJACENCY OF CLASSICAL PRETZEL KNOTS

For classical (3-strand) pretzel knots (including even type and odd type), we study their 2-adjacency using Conway polynomial and Jones polynomial. We show that only the trefoil knot and the figure-eight knot in these knots are 2-adjacent.

UNKNOTTING NUMBER AND GENUS OF 3-BRAID KNOTS

Let u(K) and g(K) denote the unknotting number and the genus of a knot K, respectively. For a 3-braid knot K, we show that u(K) ≤ g(K) holds, and that if u(K) = g(K) then K is either a 2-braid knot, a connected sum of two 2-braid knots, the figure-eight knot, a strongly quasipositive knot or its mirror image.

Introduction to Khovanov Homologies. II. Reduced Jones superpolynomials

A second part of detailed elementary introduction into Khovanov homologies. This part is devoted to reduced Jones superpolynomials. The story is still about a hypercube of resolutions of a link diagram. Each resolution is a collection of non-intersecting cycles, and one associates a 2-dimensional vector space with each cycle. Reduced superpolynomial arises when for all cycles, containing a "marked" edge of the link diagram, the vector space is reduced to 1-dimensional. The rest remains the same. Edges of the hypercube are associated with cut-and-join operators, acting on the cycles. Superpartners of these operators can be combined into differentials of a complex, and superpolynomial is the Poincare polynomial of this complex. HOMFLY polynomials are practically the same in reduced and unreduced case, but superpolynomials are essentially different, already in the simplest examples of trefoil and figure-eight knot.

The coloured Jones polynomial, the Chern-Simons invariant, and the Reidemeister torsion of the figure-eight knot

We show that from the asymptotic behaviour of an evaluation of the coloured Jones polynomial of the figure-eight knot we can extract the Chern–Simons invariant and the twisted Reidemeister torsion associated with a representation of the fundamental group of the knot complement to the two-dimensional complex special linear group.

Stable complexity and simplicial volume of manifolds

Let the Δ-complexity σ(M) of a closed manifold M be the minimal number of simplices in a triangulation of M. Such a quantity is clearly submultiplicative with respect to finite coverings, and by taking the infimum on all finite coverings of M normalized by the covering degree, we can promote σ to a multiplicative invariant, a characteristic number already considered by Milnor and Thurston, which we denote by σ∞(M) and call the stable Δ-complexity of M. We study here the relation between the stable Δ-complexity σ∞(M) of M and Gromov's simplicial volume ||M||. It is immediate to show that ||M|| ⩽ σ∞ (M) and it is natural to ask whether the two quantities coincide on aspherical manifolds with residually finite fundamental groups. We show that this is not always the case: there is a constant Cn < 1 such that ||M|| ⩽ Cnσ∞(M) for any hyperbolic manifold M of dimension n ⩾ 4. The question in dimension 3 is still open in general. We prove that σ∞(M) = ||M|| for any aspherical irreducible 3-manifold M whose JSJ decomposition consists of Seifert pieces and/or hyperbolic pieces commensurable with the figure-eight knot complement. The equality holds for all closed hyperbolic 3-manifolds if a particular three-dimensional version of the Ehrenpreis conjecture is true.