Whitehead Link Complement Research Articles

All Dehn Fillings of the Whitehead Link Complement are Tetrahedron Manifolds

In this paper we show that Dehn surgeries on the oriented components of the Whitehead link yield tetrahedron manifolds of Heegaard genus $\le 2$. As a consequence, the eight homogeneous Thurston 3-geometries are realized by tetrahedron manifolds of Heegaard genus $\le 2$. The proof is based on techniques of Combinatorial Group Theory, and geometric presentations of manifold fundamental groups.

Spherical CR uniformization of Dehn surgeries of the Whitehead link complement

We apply a spherical CR Dehn surgery theorem in order to obtain infinitely many Dehn surgeries of the Whitehead link complement that carry spherical CR structures. We consider as starting point the spherical CR uniformization of the Whitehead link complement constructed by Parker and Will, using a Ford domain in the complex hyperbolic plane $\mathbb{H}^2_{\mathbb{C}}$. We deform the Ford domain of Parker and Will in $\mathbb{H}^2_{\mathbb{C}}$ in a one parameter family. On the one side, we obtain infinitely many spherical CR uniformizations on a particular Dehn surgery on one of the cusps of the Whitehead link complement. On the other side, we obtain spherical CR uniformizations for infinitely many Dehn surgeries on the same cusp of the Whitehead link complement. These manifolds are parametrized by an integer $n \geq 4$, and the spherical CR structure obtained for $n = 4$ is the Deraux-Falbel spherical CR uniformization of the Figure Eight knot complement.

On SL(3, $${\varvec{\mathbb {C}}}$$ C )-representations of the Whitehead link group

We describe a family of representations in SL(3,C) of the fundamental group π of the Whitehead link complement. These representations are obtained by considering pairs of regular order three elements in SL(3,C) and can be seen as factorising through a quotient of π defined by a certain exceptional Dehn surgery on the Whitehead link. Our main result is that these representations form an algebraic component of the SL(3,C)-character variety of π.

A complex hyperbolic Riley slice

We study subgroups of ${\rm PU}(2,1)$ generated by two non-commuting unipotent maps $A$ and $B$ whose product $AB$ is also unipotent. We call $\mathcal{U}$ the set of conjugacy classes of such groups. We provide a set of coordinates on $\mathcal{U}$ that make it homeomorphic to $\mathbb{R}^2$ . By considering the action on complex hyperbolic space $\mathbf{H}^2_{\mathbb{C}}$ of groups in $\mathcal{U}$, we describe a two dimensional disc ${\mathcal Z}$ in $\mathcal{U}$ that parametrises a family of discrete groups. As a corollary, we give a proof of a conjecture of Schwartz for $(3,3,\infty)$-triangle groups. We also consider a particular group on the boundary of the disc ${\mathcal Z}$ where the commutator $[A,B]$ is also unipotent. We show that the boundary of the quotient orbifold associated to the latter group gives a spherical CR uniformisation of the Whitehead link complement.

Identifying the Canonical Component for the Whitehead Link

In this paper we determine topologically the canonical component of the SL2(C) character variety of the Whitehead link complement.

The minimal volume orientable hyperbolic 2-cusped 3-manifolds

We prove that the Whitehead link complement and the ( − 2 , 3 , 8 ) (-2,3,8) pretzel link complement are the minimal volume orientable hyperbolic 3-manifolds with two cusps, with volume 3.66... 3.66... = 4 × \times Catalan’s constant. We use topological arguments to establish the existence of an essential surface which provides a lower bound on volume and strong constraints on the manifolds that realize that lower bound.

Canonical triangulations of Dehn fillings

Every cusped, finite-volume hyperbolic three-manifold has a canonical decomposition into ideal polyhedra. We study the canonical decomposition of the hyperbolic manifold obtained by filling some (but not all) of the cusps with solid tori: in a broad range of cases, generic in an appropriate sense, this decomposition can be predicted from that of the unfilled manifold. As an example, we treat all hyperbolic fillings on one cusp of the Whitehead link complement.

Pants immersed in hyperbolic 3-manifolds

We show that a properly immersed thrice-punctured sphere in a cusped orientable hyperbolic 3-manifold is either embedded or has a single clasp in a manifold that is the Whitehead link complement or obtained by hyperbolic Dehn filling on a cusp of the Whitehead link complement.

Constructing spherical CR manifolds by gluing tetrahedra

We propose a general method of constructing spherical CR manifolds by gluing tetrahedra adapted to CR geometry. We obtain spherical CR structures on the complement of the figure eight knot and the Whitehead link complement with holonomy in PU ( 2 , 1 , Z [ ω ] ) and PU ( 2 , 1 , Z [ i ] ) respectively (the same integer rings appearing in real hyperbolic geometry). To cite this article: E. Falbel, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Epstein–Penner Decompositions and Thrice Punctured Spheres in Hyperbolic 3-Manifolds

Let M be a cusped hyperbolic 3-manifold containing an incompressible thrice punctured sphere S. Suppose that M is not the Whitehead link complement. We prove that a certain arc on S is isotopic to an edge of a Euclidean decomposition of M. By using the above result, we relate alternating knot diagrams and the canonical decompositions. Let K be an alternating hyperbolic knot. On a reduced alternating knot diagram of K, if we replace one of the crossings with a large number of half twists, the polar axis of the crossing is isotopic to an edge of the canonical decomposition for the resulting knot.

Surface groups in some surgered manifolds

We show that closed π 1-injective quasi-Fuchsian surfaces, immersed in a complete hyperbolic 3-manifold of finite volume, will remain π 1-injective after all but finitely many Dehn Surgeries. We use the theory of arithmetic manifolds to construct infinite families of totally geodesic surfaces in the figure-eight knot complement and the Whitehead Link complement. We use these results to show that all surgeries, except 1/0, on the figure-eight knot complement yield manifolds which contain a surface group. Furthermore, we show that all k-twist knots ( k>10) contain a closed, π 1-injective surface which will remain π 1-injective after all but at most 60 surgeries.

A combinatorial invariant for spherical CR structures

We study a cross-ratio of four generic points of $S^3$ which comes from spherical CR geometry. We construct a homomorphism from a certain group generated by generic configurations of four points in $S^3$ to the pre-Bloch group $\mathcal{P}(\mathbb{C})$. If $M$ is a 3-dimensional spherical CR manifold with a CR triangulation, by our homomorphism, we get a $\mathcal{P}(\mathbb{C})$-valued invariant for $M$. We show that when applying to it the Bloch-Wigner function, it is zero. Under some conditions on $M$, we show the invariant lies in the Bloch group $\mathcal{B}(k)$, where $k$ is the field generated by the cross-ratio. For a CR triangulation of the Whitehead link complement, we show its invariant is a torsion in $\mathcal{B}(k)$ and for a triangulation of the complement of the 52-knot we show that the invariant is not trivial and not a torsion element.