Isotopy classes of crossing arcs in hyperbolic alternating links

Transcription

1 Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic 1 altern / 21 Isotopy classes of crossing arcs in hyperbolic alternating links Anastasiia Tsvietkova Rutgers University, Newark

2 Geometry and topology of 3-manifolds: global picture Geometrization (Thurston 1976, Perelman 2006 completing Hamilton s program) prompted the study of topological objects, manifolds, using geometry. Thurston demonstrated that many 3-manifolds have hyperbolic metric or can be decomposed into pieces with such a metric. Mostow-Prasad rigidity assures that for a manifold with finite volume, the metric is unique as long as it is complete. W. Thurston demonstrated that every link in a 3-sphere is either a torus link, a satellite link, or a hyperbolic link, and these three categories are mutually exclusive. Every alternating link is either a composite link, or a hyperbolic link, or a (2, n)-torus link, and this can be seen from a link diagram (Menasco 1984). Anastasiia Tsvietkova (RutgersIsotopy University, Newark) classes of crossing arcs in hyperbolic 2altern / 21

3 Geometry and topology of 3-manifolds: local picture We still know little about the relations between intrinsic topology and intrinsic geometry of hyperbolic 3-manifolds. E.g., given an arc in a 3-manifold with a topological description, what geometric properties does it have? Some arcs with a topological description: 1) An unknotting tunnel (alternatively, a tunnel arc). A manifold that admits a single unknotting tunnel (and is not a solid torus) is called a tunnel number 1 manifold. A tunnel arc in a tunnel number 1 manifold is always homotopic to a geodesic. But when is it isotopic to a geodesic (Adams, 1995)? Conjecture (Sakuma-Weeks, 1995). Any unknotting tunnel of a hyperbolic knot is isotopic to a geodesic. Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic 3 altern / 21

4 Geometry of arcs with a topological description Some arcs that have topological description (continued): 2) A crossing arc runs from an underpass to an overpass at a crossing of a link diagram. For alternating links, the reduced alternating diagram is unique up to flypes (Tait flyping conjecture, Menasco-Thistlethwaite). Thus, being a crossing arc is a topological property: it does not depend on a diagram. Conjecture (Sakuma-Weeks, 1995). All crossing arcs in a reduced alternating diagram of a hyperbolic alternating link are isotopic to geodesics. Note: for link complements in S 3, tunnel arcs are often a subset of crossing arcs. Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic 4 altern / 21

5 What has been proved? 1) Tunnel arcs Is a tunnel arc in a tunnel number 1 manifold always isotopic to a geodesic? In a hyperbolic knot? Both questions are open. Partial progress Adams-Reid: some tunnel arcs in 2-bridge links are isotopic to geodesics. Cooper-Futer-Purcell: all tunnel arcs in certain one-cusped manifolds are isotopic to geodesics. The manifolds are all manifolds obtained by Dehn filling on one cusp of a two-cusped hyperbolic manifold, with meridian and longtitude for the filling both sufficiently large (i.e. finitely many slopes are avoided) Generalization: take a collection of arcs in a manifold such that the complement is a collection of handlebodies (and arcs can be isotoped fixing endpoints to be disjoint). Burton-Purcell: some evidence that collections of n such arcs may not be isotopic to geodesics in tunnel number n manifolds. Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic 5 altern / 21

6 What has been proved? 2) Crossings arcs Conjecture (Sakuma-Weeks, 1995). All crossing arcs in a reduced alternating diagram of a hyperbolic alternating links are isotopic to geodesics. Proved for: 2-bridge links (Gueritad-Futer, 2006, as well as an unpublished proof by Akiyoshi-Sakuma-Wada) Three infinite families of links with highly symmetric diagrams (Sakuma and Weeks, 1995): 3-braids (σ 1 σ 1 2 )n for n > 1; n-component chains for n > 2; and links C (n) on the right for n even and n > 3. Several more links by Aitchison-Reeves and Hatcher. Overall, 4 infinite families, and few more (up to 20) links outside these families. Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic 6 altern / 21

7 Hyperbolic 2-bridge links (Gueritad); hyperbolic 1-punctured torus bundles (Lackenby); three symmetric families of links (mentioned above, Salkuma-Weeks). No other classes of 3-manifolds for which the CCD is even conjecturally understood. Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic 7 altern / 21 Related questions and significance Canonical cell decomposition (CCD) Epstein-Penner, 1988: every hyperbolic 3-manifold has a decomposition into hyperbolic convex polyhedra. This decomposition is a complete topological invariant of hyperbolic 3-manifolds. Arcs of a canonical cell decomposition are isotopic to geodesics.

8 Related questions and significance (continued) Ideal geodesic triangulation - a triangulation that agrees with the complete hyperbolic structure of the 3-manifold M. Ideal means that vertices are on the torus boundary of M. Edges of an ideal geodesic triangulation are isotopic to geodesics. There is no algorithm known that constructs such a triangulation for an arbitrary hyperbolic 3-manifold with torus boundary. Even for hyperbolic links, this is an open question (solved by Gueritad and Futer for 2-bridge links). Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic 8 altern / 21

9 Related questions and significance (continued) Length of arcs. For a hyperbolic geodesic, one can measure its hyperbolic length. Empirically, for certain classes of manifolds there are upper and lower bounds for the length, and for the related cusp area. Conjecture (Thistlethwaite). Crossing arcs of alternating links are short. In particular, if the meridian of a link fixed to be 1, the hyperbolic intercusp length is at most log 8. Lackenby and Purcell recently offered another fact supporting the conjecture. The conjecture however is still open. Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic 9 altern / 21

10 (A.T.) Sufficient conditions for crossings arcs in alternating links to be isotopic to geodesics. The conditions are numerical (a set of simple inequalities for parameters that can be computed from a link diagram). As a result: a) New examples of links, including infinite families, with crossings arcs isotopic to geodesics. E.g., the family of closed braids with the braid word (σ 1 σ 3 σ2 1 )n, k > 0, n > 1. One can use similar argument for any of the infinitely many families (σ 1 σ 3...σ 2k±1 σ2 1 σ σ 1 2k )n, k > 0, n > 1. One of Sakuma-Weeks three families, braids (σ 1 σ2 1 )n, is a subset. b) Many triangulations are ideal geodesic triangulations. The links are then hyperbolic without a reference to Geometrization. Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic10 altern / 21

11 Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic11 altern / 21 Proof: main ingridients I. (Thislethwaite-T.) An alternative method for computing hyperbolic structure of links: assigns labels with geometric meaning to a link diagram.

12 Proof: main ingridients I. (Thislethwaite-T.) An alternative method for computing hyperbolic structure of links: assigns labels with geometric meaning to a link diagram. In the upper half-space model of H 3, a ball tangent to the boundary of H 3 is a horoball, bounded by a horosphere. In H 3, the preimage of a boundary torus for a hyperbolic link is a set of horospheres. A meridian is a simple closed curve traveling once around the torus. Its preimage is on a horosphere. Parameterize Euclidean translations on each horosphere by complex numbers so that the meridional translation corresponds to 1. If one of the horospheres is the plane z = 1, the horoballs have disjoint interiors (follows from Adams results). nastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic11 altern / 21

13 A crossing label contains the information about the distance and angle between the strands of the link. In particular, the red (geodesic) arc has γ as the preimage in H 3, joining the centers of the corresponding horospheres. The modulus of the label determines the hyperbolic (cusp-to-cusp) distance along the arc, and its argument is the angle between the meridional translations on horospheres. An edge label contains information about a preimage of an arc traveling on the boundary torus between two crossings. In H 3, it is a complex number determining the Euclidean translation along β. The arc β travels between the points where the preimages of crossings arcs pierce the horosphere. Its orientation agrees with the orientation of the link. Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic12 altern / 21

14 Proof: main ingridients Using isometries rotating the regions of the diagram, one obtains a system of polynomial equations in edge and crossing labels. One of the solutions describes the complete hyperbolic structure. For our proof: rather than starting with a hyperbolic link, take an arbitrary alternating link diagram and assign labels (possibly, with no geometric meaning). Want to obtain the conditions on the labels that guarantee the existence of a complete hyperbolic structure. Compare with completeness and consistency conditions on a triangulation by W. Thurston. Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic13 altern / 21

15 Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic14 altern / 21 Proof: main ingridients II. Thurston-Menasco s decomposition: 2 polyhedra, above and below a link diagram, mirror images of each other. Correspond to the combinatorics of the diagram closely. To construct one, place an ideal vertex on every edge of the diagram (i.e. on a segment from a crossing to the subsequent crossing). For every crossing of the diagram, place 4 polyhedral edges around it, 2 for the polyhedron above, and 2 for the one below (these will be identified later into just one edge). Identify two edges of every bigon of the diagram so that there are no more bigons.

16 Proof: main ingridients II. Thurston-Menasco s decomposition: 2 polyhedra, above and below a link diagram, mirror images of each other. Correspond to the combinatorics of the diagram closely. To construct one, place an ideal vertex on every edge of the diagram (i.e. on a segment from a crossing to the subsequent crossing). For every crossing of the diagram, place 4 polyhedral edges around it, 2 for the polyhedron above, and 2 for the one below (these will be identified later into just one edge). Identify two edges of every bigon of the diagram so that there are no more bigons. After identifications, every n-sided region of the diagram that is not a bigon yields an n-sided face of a polyhedron. The edges of the polyhedra are crossing arcs. Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic14 altern / 21

17 Proof: main idgridients II. Thurston-Menasco s polyhedra are polyhedra only in topological sense: the faces might not be planar. Every ideal vertex corresponds to a triangular or rectangular cross-section. if the link is hyperbolic, the cross-sections are Euclidean polygons. For our proof: decompose the link complement into two (possibly non-geometric) polyhedra. Using the labels, track what happens with cross-sections of the polyhedra. Angles of the cross-sections are captured by ratios of two corresponding edge labels, e.g. Im u v + 1. Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic15 altern / 21

18 Proof: main ingridiients III. (Petronio-Weeks + Thurston) A topological triangulation is an ideal geodesic triangulation if and only if 1) Every simplex is a hyperbolic tetrahedron (three pairs of opposite edges correspond to the arguments of z, 1 1/z, 1/(1 z) for a complex number z, called a tetrahedral shape or parameter). 2) At every edge of a triangulation, after faces of n tetrahedra are identified in pairs and glued at that edge to obtain the manifold, the corresponding shapes satisfy z 1 z 2...z n = 1. 3) All tetrahedra have non-negative volume, and not all have 0-volume. 4) The metric is complete, i.e. the cross-sectional triangles are glued together to give a closed Euclidean surface. Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic16 altern / 21

19 For our proof: triangulate the polyhedra. There are many ways to do this, and crossing arcs are always some of the edges. Keep track of what happens with the cross-sections using edge and crossing labels. Use the above criterion to obtain the conditions on the labels that are sufficient for the existence of a complete hyperbolic structure. Additionally, the crossing arcs are the arcs of an ideal geodesic triangulation, and are therefore isotopic to geodesics. Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic17 altern / 21

20 Geometric interpretation of the sufficient conditions Suppose P is an ideal cell in H 3, and every vertex has valency 3 or 4. Call P cross-sectionally convex if at every vertex, a Euclidean cross-section is a convex polygon. The conditions we obtained force Menasco-Thurston polyhedra to be cross-sectionally convex and not to have cusp-to-cusp edges of length 0. We prove that once faces of such a complex are triangulated, the resulting polyhedron is properly embedded in H 3. Equivalently, the Menasco-Thurston decomposition of a link complement, with faces triangulated, can be realized by ideal hyperbolic polyhedra. Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic18 altern / 21

21 Anastasiia Tsvietkova (Rutgers Isotopy University, Newark) classes of crossing arcs in hyperbolic19 altern / 21 Conjecture. For a hyperbolic alternating link, the preimage of every Menasco-Thurston polyhedron is cross-sectionally convex. The polyhedra can then be subdivided into partially flat ideal geodesic triangulation. Question. For which links, in addition, the faces of these polyhedra are planar (i.e. lie in one hyperbolic plane)? If the polyhedra are cross-sectionally convex, and faces are planar, the resulting decomposition is the canonical cell decomposition. This happens, for example, for the link (σ 1 σ 3 σ 1 2 )4.

22 Infinite families of closed braids To check that sufficient conditions hold for the closed braids 1 n (σ1 σ3...σ2k+1 σ2 1 σ4 1...σ2k ), use symmetry of the diagram. The central region corresponds to a regular ideal polygon. Labels are obtained by a computation performed for an infinite family at once. Since a triangulation might be partially flat, check that edges are simple geodesics. Anastasiia Tsvietkova (RutgersIsotopy University, Newark) classes of crossing arcs in hyperbolic 20altern / 21

23 Infinite families of closed braids To check that sufficient conditions hold for the closed braids 1 n (σ1 σ3...σ2k+1 σ2 1 σ4 1...σ2k ), use symmetry of the diagram. The central region corresponds to a regular ideal polygon. Labels are obtained by a computation performed for an infinite family at once. Since a triangulation might be partially flat, check that edges are simple geodesics. Acknowledgments: the work was motivated by discussions with Marc Lackenby. Anastasiia Tsvietkova (RutgersIsotopy University, Newark) classes of crossing arcs in hyperbolic 20altern / 21

24 Okinawa Institute of Science and Technology (OIST), Japan, is currently inviting applications in topology and geometry of manifolds for postdoctoral positions and positions for senior researchers willing to spend at least 6 months in residence (no teaching duties): Anastasiia Tsvietkova (RutgersIsotopy University, Newark) classes of crossing arcs in hyperbolic 21altern / 21