Exact Volume of Hyperbolic 2-bridge Links

Transcription

1 Exact Volume of Hyperbolic 2-bridge Links Anastasiia Tsvietkova Louisiana State University Parts of this work are joint with M. Thislethwaite, O. Dasbach Anastasiia Tsvietkova (Louisiana State University Exact Parts Volume of this of work Hyperbolic are joint 2-bridge with M. Links Thislethwaite, O. Dasbach) 1 / 17

2 Motivation: exact hyperbolic volume As a corollary of W. Thurston s Hyperbolization Theorem, many 3 manifolds have a hyperbolic metric or can be decomposed into pieces with hyperbolic metric. In view of Mostow-Prasad rigidity, such metric is unique as long as it is complete for a manifold with finite volume. The volume itself is one of the fundamental topological invariants of hyperbolic 3-manifolds. W. Thurston suggested a method for computing the volume, based on a triangulation of a manifold. It was implemented in SnapPea (J. Weeks), which produces a decimal approximation as a result. The program Snap (O. Goodman, C. Hodgson, W. Neumann) followed. It approximates the hyperbolic structure to a high precision, and then makes an intelligent guess of the corresponding algebraic numbers, from which the volume can be computed. We want to construct such a polynomial, that the volume can be expressed as an analytic function of one of its roots. Anastasiia Tsvietkova (Louisiana State University Exact Parts Volume of this of work Hyperbolic are joint 2-bridge with M. Links Thislethwaite, O. Dasbach) 2 / 17

3 Motivation: relating volume to a link diagram Another motivating question concerns relating a diagram of a link to the geometry of its complement and, in particular, to its hyperbolic volume. M. Lackenby showed that the hyperbolic volume of an alternating link has bounds as functions of the number of twists of a reduced, alternating diagram. D. Futer, E. Kalfagianni and J. Purcell extended these results to some other classes of links. The suggested computation of the exact volume is based solely on the layout of a reduced link diagram Anastasiia Tsvietkova (Louisiana State University Exact Parts Volume of this of work Hyperbolic are joint 2-bridge with M. Links Thislethwaite, O. Dasbach) 3 / 17

4 Methods for describing hyperbolic structure A well-known method for describing the structure of hyperbolic manifolds by W. Thurston is based on decomposition of the complement into ideal tetrahedra. An alternative method (with M. Thistlethwaite) is based on ideal polygons bounding the regions of a diagram of the link. It is applicable to all hyperbolic links whose diagrams satisfy a few mild restrictions. In particular, it is applicable to hyperbolic alternating links (due to results by D. Futer, E. Kalfagianni, J. Purcell). Anastasiia Tsvietkova (Louisiana State University Exact Parts Volume of this of work Hyperbolic are joint 2-bridge with M. Links Thislethwaite, O. Dasbach) 4 / 17

5 Horoball packing The preimage of a cusp of a link complement in H 3 is a set of horoballs. Cusps may be chosen so that the horoballs have disjoint interiors. The alternative method parameterizes a horoball pattern using complex labels, assigned to crossing and edges of the link diagram. The labels contain the information about hyperbolic structure and determine a representation of a link group into PSL 2 (C). Anastasiia Tsvietkova (Louisiana State University Exact Parts Volume of this of work Hyperbolic are joint 2-bridge with M. Links Thislethwaite, O. Dasbach) 5 / 17

6 An alternative method: preliminaries Take horospherical cross-sections of the cusps, so that length of a meridian on each cross-section is 1. Then cross-sectional tori from distinct cusps are disjoint, and each torus is embedded in the link complement. Exception: figure-eight knot complement, where the cross-sectional torus touches itself. (C. Adams). Specify a complex affine structure on each horosphere (so that meridian corresponds to 1), and choose the coordinates in H 3 (so that one of the horospheres is the Euclidean plane z=1). Anastasiia Tsvietkova (Louisiana State University Exact Parts Volume of this of work Hyperbolic are joint 2-bridge with M. Links Thislethwaite, O. Dasbach) 6 / 17

7 An alternative method: edge and crossing labels A k-sided region in the link diagram is a disk whose boundary is a union of k arcs on the boundary torus and k arcs at crossings. The preimage of this boundary in H 3 determines a cyclic sequence of horospheres H 1,..., H k. A crossing label contains the geometric information about a preimage of the corresponding crossing arc from an overpass to an underpass. An edge label contains information about a preimage of the corresponding arc travelling between adjacent crossings on the boundary torus (it is a complex number determining a translation along β j ). Anastasiia Tsvietkova (Louisiana State University Exact Parts Volume of this of work Hyperbolic are joint 2-bridge with M. Links Thislethwaite, O. Dasbach) 7 / 17

8 An alternative method: relations To find the labels, we use isometries of ideal polygons arising from the regions of a link diagram. In particular, we compose the isometries rotating the region polygons. Since every k-sided polygon closes up, the composite of k such isometries is 1. Represent the Möbius transformations by 2 2 matrices. From the matrix entries we read off three independent polynomial relations for every region in edge and crossing labels. One can write a combinatorial formula for the relations, that depends only on the number of sides in a region. Anastasiia Tsvietkova (Louisiana State University Exact Parts Volume of this of work Hyperbolic are joint 2-bridge with M. Links Thislethwaite, O. Dasbach) 8 / 17

9 Example: the Borromean Rings w 1 u 2 1 = w 1 u 2 2 = w 1 (u 1 +1)(u 2 +1) = w 2 (u 1 +1) 2 = w 1 (u 1 +1) 2 = w 2 u 1 u 3 = 1. Hence, u 1 = u 2 = u 3 = 1 2 ( 1 + i), w 1 = i 2 = w 2. Anastasiia Tsvietkova (Louisiana State University Exact Parts Volume of this of work Hyperbolic are joint 2-bridge with M. Links Thislethwaite, O. Dasbach) 9 / 17

10 Canonical cell decomposition of hyperbolic 2-bridge links Consider a hyperbolic 2-bridge link with n crossings. According to the Sakuma-Weeks description of the canonical cell decomposition, there are 2(n 3) tetrahedra occurring in isometric pairs. Picture by J. Weeks The red dotted lines represent geodesics in the link complement that correspond to edges of the canonical cell decomposition. Anastasiia Tsvietkova (Louisiana State University Exact Parts Volume of this of work Hyperbolic are joint 2-bridge with M. Links Thislethwaite, O. Dasbach) 10 / 17

11 Volume from shapes of tetrahedra The figure on the left illustrates a Euclidean cusp cross-section of an ideal tetrahedron. Suppose that the (complex) translations corresponding to the sides of the triangular cross-section are u, v, (u + v). Any of v u, u u+v, u+v v can be taken as a shape z of the ideal tetrahedron. The volume of the tetrahedron is then computed using Lobachevsky function as Λ(arg(z)) + Λ(arg( 1 1 z )) + Λ(arg(1 1 z )). Anastasiia Tsvietkova (Louisiana State University Exact Parts Volume of this of work Hyperbolic are joint 2-bridge with M. Links Thislethwaite, O. Dasbach) 11 / 17

12 Finding tetrahedra shapes from a link diagram Let the strands of a standard two-bridged link diagram be s 1, s 2, s 3, s 4. For every pair of isometric tetrahedra, there is a Euclidean cross-section with vertices v 1, v 2, v 3 on one of (thickened) strands s i. The cross-section has its three vertices on geodesics joining s i with each of the other three strands. Anastasiia Tsvietkova (Louisiana State University Exact Parts Volume of this of work Hyperbolic are joint 2-bridge with M. Links Thislethwaite, O. Dasbach) 12 / 17

13 Finding shapes and the polynomial The shapes of consecutive pairs of isometric tetrahedra can be written in terms of edge labels as bj i. In their turn, all edge labels in can be b j i+1 written recursively in terms of one crossing label w 1. The last region of the link diagram gives a polynomial in w 1. The formulas for the labels and the polynomial are a bit unwieldy, but are easy to use, and are obtained in a straightforward manner from the region relations. Anastasiia Tsvietkova (Louisiana State University Exact Parts Volume of this of work Hyperbolic are joint 2-bridge with M. Links Thislethwaite, O. Dasbach) 13 / 17

14 Example: twist knot with k + 2 crossings There are k 1 pairs of isometric tetrahedra. The shape parameters are the ratios z i = b i b i 1. All the labels can be found recursively from the crossing label w as b 0 = ( 1) k w, b i+1 = 1 + w 2 b i. The relation b k = 0 gives a polynomial in w. Anastasiia Tsvietkova (Louisiana State University Exact Parts Volume of this of work Hyperbolic are joint 2-bridge with M. Links Thislethwaite, O. Dasbach) 14 / 17

15 Exact volume The closed combinatorial formulas can be obtained as well. E. g. for a twist knot with 2n + 2 crossings (with O. Dasbach, LSU VIGRE student group), the polynomial in w is n j=0 ( 2n j j ) w 2j + n 1 j=0 ( 2n j 1 ) k w 2j+1 = 0. The polynomial has several roots. The root with the greatest volume is the geometric one. Similar techniques and formulas by C. Zickert can be used to compute the complex volume of hyperbolic 2-bridged links (its real part is hyperbolic volume, and the imaginary part is the Chern-Simons invariant). Anastasiia Tsvietkova (Louisiana State University Exact Parts Volume of this of work Hyperbolic are joint 2-bridge with M. Links Thislethwaite, O. Dasbach) 15 / 17

16 Example: (3, 3) link There are 3 pairs of isometric tetrahedra. The polynomial and the root: (u 2 + u 1)(u 2 + u + 1) = 1, u = The exact volume is 12Λ( π) + 6Λ( 2π) = 8Λ( π ), which is twice the volume of the figure eight knot complement. Anastasiia Tsvietkova (Louisiana State University Exact Parts Volume of this of work Hyperbolic are joint 2-bridge with M. Links Thislethwaite, O. Dasbach) 16 / 17

17 Questions Horoball packing of Turk s Head knot. Pictures of horoball packings by M. Thistlethwaite Anastasiia Tsvietkova (Louisiana State University Exact Parts Volume of this of work Hyperbolic are joint 2-bridge with M. Links Thislethwaite, O. Dasbach) 17 / 17