It is known that the shortest non-simple closed geodesic on an orientable hyperbolic 2-orbifold passes through an orbifold point of the orbifold. This raises questions about minimal length non-simple closed geodesics disjoint from the orbifold points. Here we explore once self-intersecting closed geodesics disjoint from the orbifold points of the orbifold, called figure eight geodesics. Using fundamental domains and basic hyperbolic trigonometry we identify and classify all figure eight geodesics on triangle group orbifolds. This classification allows us to find the shortest figure eight geodesic on a triangle group orbifold, namely the unique one on the (3,3,4)-triangle group orbifold. We then show that this same curve is the shortest figure eight geodesic on a hyperbolic 2-orbifold without orbifold points of order two.
In this paper, we compute the symplectic Floer homology of the figure eight knot. This provides first nontrivial knot with trivial symplectic Floer homology.
Using Dowlin's spectral sequence from Khovanov homology to knot Floer homology, we prove that reduced Khovanov homology (over $\mathbb{Q}$) detects the figure-eight knot.
We show that most cabled knots over the figure eight knot in $S^3$ satisfy the AJ-conjecture, in particular, any $(r,s)$-cabled knot over the figure eight knot satisfies the $AJ$-conjecture if $r$ is not a number between $-4s$ and $4s$.
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.
This paper determines the minimal degree sequence for two compact rational knots, namely the trefoil and figure-eight knots. We find explicit projections with the minimal degree sequence of each knot. This is done by modifying a non-compact rational minimal-degree parameterization of the trefoil and figure-eight knots to make it compact. This research was conducted at Ramapo College of New Jersey during the summer of 2011 with Dr. Donovan McFeron.
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.
We obtain a branched spherical CR structure on the complement of the figure eight knot with a given holonomy representation (called rho_2). There are essentially two boundary unipotent representations from the complement of the figure eight knot into PU(2,1), we call them rho_1 and rho_2. 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 behaviour of of the fundamental group of the fiber under the representation is a key difference between rho_1 and rho_2.
We will study the asymptotic behaviors of the colored Jones polynomials of the figure-eight knot. In particular we will show that for certain limits we obtain the volumes of the cone manifolds with singularities along the knot.
We show some computations on representations of the fundamental group in SL(2;C) and Reidemeister torsion for a homology 3-sphere obtained by Dehn surgery along the figure-eight knot. This is the second version. We recorrected several errors in the first version.
We calculate limits of the colored Jones polynomials of the figure-eight knot and conclude that in most cases they determine the volumes and the Chern--Simons invariants of the three-manifolds obtained by Dehn surgeries along it.
We establish an upper bound $ω(p/q)$ on the complexity of manifolds obtained by $p/q$-surgeries on the figure eight knot. It turns out that if $ω(p/q)\leqslant 12$, the bound is sharp.
The paper contains the computation of the noncommutative A-ideal of the figure-eight knot, a noncommutative generalization of the A-polynomial. We show that if a knot has the same noncommutative A-ideal as the figure-eight knot, then all colored Kauffman brackets are the same as those of the figure-eight knot.
This paper considers "geometric" ideal triangulations of cusped hyperbolic 3-manifolds, i.e. decompositions into positive volume ideal hyperbolic tetrahedra. We exhibit infinitely many geometric ideal triangulations of the figure eight knot complement. As far as we know, this is the first construction of infinitely many geometric triangulations of a cusped hyperbolic 3-manifold. In contrast, our approach does not extend to the figure eight sister manifold, and it is unknown if there are infinitely many geometric triangulations for this manifold.
Let M be a closed 3-manifold obtained by Dehn surgery along the figure-eight knot. We give a formula of the Reisdemeisiter torsion of M for any irreducible SL(2;C)-representation. It is described as a rational expression of the trace of the image of the meridian.
In this paper we find infinitely many lattices in $SL(4,\mathbb{R})$ each of which contains thin subgroups commensurable with the figure-eight knot group.
We show that from the asymptotic behavior of an evaluation of the colored 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.
In this paper we investigate the asymptotic behavior of the colored HOMFLY polynomial of the figure eight knot associated with the symmetric representation. We establish an analogous asymptotic expansion for the colored HOMFLY polynomial. From the asymptotic behavior we show that the Chern-Simons invariants and twisted Reidemeister torsion can be obtained with suitable modification of the case of colored Jones polynomial.
We show that all positive contact surgeries on every Legendrian figure-eight knot in $(S^3, ξ_{\rm{std}})$ result in an overtwisted contact structure. The proof uses convex surface theory and invariants from Heegaard Floer homology.
The volume conjecture and its generalization state that the series of certain evaluations of the colored Jones polynomials of a knot would grow exponentially and its growth rate would be related to the volume of a three-manifold obtained by Dehn surgery along the knot. In this paper, we show that for the figure-eight knot the series converges in some cases and the limit equals the inverse of its Alexander polynomial.
We give a rigorous proof of the colored HOMFLY-PT polynomials of the trefoil knot, the figure-eight knot and twist knots. For the trefoil knot and the figure-eight knot, it is expressed by a single sum, and for a twist knot, it is expressed by a double sum.
In earlier work, the authors introduced a conjecture which, for an orientation-preserving diffeomorphism $φ\colon S \to S$ of a surface, connects a certain quantum invariant of $φ$ with the hyperbolic volume of its mapping torus $M_φ$. This article provides a proof of this conjecture in the simplest case where it applies, namely when the surface $S$ is the one-puncture torus and the mapping torus $M_φ$ is the complement of the figure-eight knot.
We describe a simple fundamental domain for the holonomy group of the boundary unipotent spherical CR uniformization of the figure eight knot complement, and deduce that small deformations of that holonomy group (such that the boundary holonomy remains parabolic) also give a uniformization of the figure eight knot complement. Finally, we construct an explicit 1-parameter family of deformations of the boundary unipotent holonomy group such that the boundary holonomy is twist-parabolic. For small values of the twist of these parabolic elements, this produces a 1-parameter family of pairwise non-…
We give a Dehn surgery characterization of the trefoil and the figure eight knots. These results are gotten by combining surgery formulas in Heegaard Floer homology from an earlier paper with the characterization of these knots in terms of their knot Floer homology given in a recent paper of Ghiggini.
In this paper we investigate the asymptotic behavior of the colored Jones polynomials and the Turaev-Viro invariants for the figure eight knot. More precisely, we consider the $M$-th colored Jones polynomials evaluated at $(N+1/2)$-th root of unity with a fixed limiting ratio, $s$, of $M$ and $(N+1/2)$. We find out the asymptotic expansion formula (AEF) of the colored Jones polynomials of the figure eight knot with $s$ close to $1$. Nonetheless, we show that the exponential growth rate of the colored Jones polynomials of the figure eight knot with $s$ close to $1/2$ is strictly less than those…
I calculate optimistically asymptotic behaviors of the WRT SU(2) invariants for the three-manifolds obtained from the figure-eight knot by p-surgeries with p=0,1,2,...,10, from which one can extract volumes and the Chern-Simons invariants of these closed manifolds. I conjecture that this also holds for general closed three-manifolds.
The $0$-surgeries of two knots $K_1$ and $K_2$ are homology cobordant rel meridians if there exists a $\mathbb{Z}$-homology cobordism $X$ between them such that the two knot meridians are in the same homology class in $H_{1}(X,\mathbb{Z})$. In this paper, we give a pair of rationally slice knots which are not smoothly concordant but whose $0$-surgeries are homology cobordant rel meridians. One knot in the pair is the figure eight knot, which has concordance order two; all previous examples of such pairs of knots are infinite order.
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 th…
We give a description of several representation varieties of the fundamental group of the complement of the figure eight knot in PGL(3,C) or SL(3,C). We moreover obtain an explicit parametrization of matrices generating the representation and a description of the projection of the representation variety into the character variety of the boundary torus into SL(3,C).
We give explicit equations that describe the character variety of the figure eight knot for the groups SL(3,C), GL(3,C) and PGL(3,C). This has five components of dimension 2, one consisting of totally reducible representations, another one consisting of partially reducible representations, and three components of irreducible representations. Of these, one is distinguished as it contains the curve of irreducible representations coming from $Sym^2:SL(2,C) \to SL(3,C)$. The other two components are induced by exceptional Dehn fillings of the figure eight knot. We also describe the action of the s…
We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of a cable of the figure-eight knot, evaluated at $\exp(ξ/N)$ for a real number $ξ$. We show that if $ξ$ is sufficiently large, the colored Jones polynomial grows exponentially when $N$ goes to the infinity. Moreover the growth rate is related to the Chern-Simons invariant of the knot exterior associated with an $\mathrm{SL}(2;\mathbb{R})$ representation.
Two of the basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. We present the first such classification on an infinite family of (mostly) hyperbolic 3-manifolds: surgeries on the figure-eight knot. We also determine which of the tight contact structures are symplectically fillable and which are universally tight.
Motivated by the theory of quantum A-ideals of Frohman-Gelca-LoFaro, the theory of q-holonomicity of quantum invariants of Garoufalidis-Le and the AJ conjecture of Garoufalidis, Sikora formulated the strong AJ conjecture which relates the A-ideal and recurrence ideal of a knot in the 3-sphere. This conjecture has been verified for all torus knots and most of their cables. In this paper, we verify the strong AJ conjecture for the figure eight knot.
Using Ohtsuki's method, we prove the Asymptotic Expansion Conjecture and the Volume Conjecture of the Reshetikhin-Turaev and the Turev-Viro invariants for all hyperbolic $3$-manifolds obtained by doing a Dehn-surgery along the figure-$8$ knot.
We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial evaluated at $\exp(ξ/N)$ for a real number $ξ$ greater than a certain constant. We prove that, from the asymptotic behavior, we can extract the $\rm{SL}(2;\mathbb{C})$ Chern--Simons invariant and the Reidemeister torsion twisted by the adjoint action both associated with a representation determined by $ξ$.
Differential expansion (DE) for a Wilson loop average in representation $R$ is built to respect degenerations of representations for small groups. At the same time it behaves nicely under some changes of the loop, e.g. of some knots in the case of $3d$ Chern-Simons theory. Especially simple is the relation between the DE for the trefoil $3_1$ and for the figure eight knot $4_1$. Since arbitrary colored HOMFLY for the trefoil are known from the Rosso-Jones formula, it is therefore enough to find their DE in order to make a conjecture for the figure eight. We fulfil this program for all rectangu…
The AJ conjecture relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been verified for some classes of knots, including all torus knots, most double twist knots, (-2,3,6n \pm 1)-pretzel knots, and most cabled knots over torus knots. In this paper we study the AJ conjecture for (r,2)-cables of a knot, where r is an odd integer. In particular, we show that the AJ conjecture holds true for (r,2)-cables of the figure eight knot, where r is an odd integer satisfying |r| \ge 9.
We establish results concerning the profinite completions of 3-manifold groups. In particular, we prove that the complement of the figure-eight knot $S^3-K$ is distinguished from all other compact 3-manifolds by the set of finite quotients of its fundamental group. In addition, we show that if $M$ is a compact 3-manifold with $b_1(M)=1$, and $π_1(M)$ has the same finite quotients as a free-by-cyclic group $F_r\rtimes\mathbb{Z}$, then $M$ has non-empty boundary, fibres over the circle with compact fibre, and $π_1(M)\cong F_r\rtimes_ψ\mathbb{Z}$ for some $ψ\in{\rm{Out}}(F_r)$.
For each natural number n >= 4, we determine the unique lowest volume hyperbolic 3-orbifold whose torsion orders are bounded below by n. This lowest volume orbifold has base space the 3-sphere and singular locus the figure-8 knot, marked n. We apply this result to give sharp lower bounds on the volume of a hyperbolic manifold in terms of the order of elements in its symmetry group.
We show that some hyperbolic 3-manifolds which are tessellated by copies of the regular ideal hyperbolic tetrahedron embed geodesically in a complete, finite volume, hyperbolic 4-manifold. This allows us to prove that the complement of the figure-eight knot geometrically bounds a complete, finite volume hyperbolic 4-manifold. This the first example of geometrically bounding hyperbolic knot complement and, amongst known examples of geometrically bounding manifolds, the one with the smallest volume.
For any unoriented loop on a compact connected oriented surface with one boundary component, the generalized Dehn twist along the loop is defined as an automorphism of the completed group ring of the fundamental group of the surface. If the loop is simple, this is the usual right handed Dehn twist, in particular realized as a mapping class of the surface. We investigate the case when the loop has a single transverse double point, and show that in this case the generalized Dehn twist is not realized as a mapping class.
The path closure of figure eight knot complement space, $N_0$, supports a natural DA (derived from Anosov) expanding attractor. Using this attractor, Franks-Williams constructed the first example of non-transitive Anosov flow on the manifold $M_0$ obtained by gluing two copies of $N_0$ through identity map along their boundaries, named by Franks-Williams manifold. In this paper, our main goal is to classify expanding attractors on $N_0$ and non-transitive Anosov flows on $M_0$. We prove that, up to orbit-equivalence, the DA expanding attractor is the unique expanding attractor supported by $N_…
We classify Legendrian torus knots and figure eight knots in the tight contact structure on the 3-sphere up to Legendrian isotopy. As a corollary to this we also obtain the classification of transversal torus knots and figure eight knots up to transversal isotopy.
The purpose of this paper is to classify Anosov flows on the 3-manifolds obtained by Dehn surgeries on the figure-eight knot. This set of 3-manifolds is denoted by M(r) (r is a ratioanl number), which contains the first class of hyperbolic 3-manifolds admitting Anosov flows in history, discovered by Goodman. Combining with the classification of Anosov flows on the sol-manifold M(0) due to Plante, we have: 1. if r is an integer, up to topological equivalence, M(r) exactly carries a unique Anosov flow, which is constructed by Goodman by doing a Dehn-Fried-Goodman surgery on a suspension Anosov f…
Explicit answer is given for the HOMFLY polynomial of the figure eight knot $4_1$ in arbitrary symmetric representation R=[p]. It generalizes the old answers for p=1 and 2 and the recently derived results for p=3,4, which are fully consistent with the Ooguri-Vafa conjecture. The answer can be considered as a quantization of the σ_R = σ_{[1]}^{|R|} identity for the "special" polynomials (they define the leading asymptotics of HOMFLY at q=1), and arises in a form, convenient for comparison with the representation of the Jones polynomials as sums of dilogarithm ratios. In particular, we construct…
We describe the explicit form and the hidden structure of the answer for the HOMFLY polynomial for the figure eight 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 eight knot, which can be also partly used in more complicated situations. Finally we report the result of exact calculation for figure eight and some other 3-strand knots based on the previously develop…
