Hyperbolic geometry on the figureeight knot complement alexander j. There are polynomialtime algorithms for the computation of the alexander polynomial of a. Used in the proof of the hyperbolic dehn lling theorem, hyperbolic cone manifolds, also for degeneration of structures, and many other situations. Knot and link complements enjoy a geometry of crystalline beauty, rigid enough that simple cutandpaste techniques meet geometrical as well as topological needs, yet surprisingly complex in their inexhaustible variety. For example in hyperbolic geometry, the sum of angles in a triangle is less than 180 degrees. Computation of hyperbolic structures in knot theory 461. Biarcs, global radius of curvature, and the computation of ideal. In this section we bring together some of the basic facts from knot theory we.
Although the subject matter of knot theory is familiar. Weeks computation of hyperbolic structures in knot theory 1. Theory of computation automata notes pdf ppt download. The completion of hyperbolic threemanifolds obtained from ideal polyhedra. Hyperbolic structures have found applications in knot tabulation 3 and more generally provide a fast and effective way to test hyperbolic knots and links for equivalence 4 and to compute their symmetry groups 5,6 and other invariants 7. Hence, one can compute an upper bound on the number. The method is to first decompose the knot or link complement into ideal tetrahedra. Many knot invariants are known and can be used to distinguish knots. The figure eight knot complement is a fiber bundle over the circle with fiber a oncepunctured torus. Computation of hyperbolic structures in knot theory. This is a survey of the impact of thurstons work on knot theory, laying emphasis on the two characteristic features, rigidity and flexibility, of 3dimensional hyperbolic structures. Hyperbolic knot theory this book is an introduction to hyperbolic geometry in dimension three, and its applications to knot theory and to geometric problems arising in knot theory. The invariant of a link in threesphere, associated with the cyclic quantum dilogarithm, depends on a natural number n.
In topology, knot theory is the study of mathematical knots. Ams proceedings of the american mathematical society. A new method for computing the hyperbolic structure of the complement of a. Thus a hyperbolic structure on a knot complement is a complete invariant of the knot. In this chapter, we wish to compute explicit complete hyperbolic structures on 3manifolds, again with our primary examples being knot complements. Representations have been used to distinguish knots, in.
An algorithm to find a finitevolume hyperbolic structure on a 3manifold, provided. In mathematics, hyperbolic geometry also called lobachevskian geometry or bolyailobachevskian geometry is a noneuclidean geometry. Although the theory of manifolds developed rapidly in the following generations,thisconjectureremainedopen. Cullershalen theory of surfaces associated to ideal points. Surgery rank 1 boundary unipotent geometric structures on the figure eight knot complement icerm workshop exotic geometric structures martin deraux institut fourier grenoble sep 16, 20. Examples are euclidean, spherical and hyperbolic geometries. Canonical triangulations provide an algorithm to compute the symmetry groups of hyperbolic links and their complements, along with information on how each symmetry acts on the link components. For any given line r and point p not on r, in the plane containing both line r and point p there are at least two distinct lines through p that do not intersect r. The program snappea 61 constructs hyperbolic structures on knot and link compliments, as well as the hyperbolic dehn surgeries on these compliments. Albert einsteins special theory of relativity is based on hyperbolic geometry. Retrieve articles in proceedings of the american mathematical society with msc 2010. Trigonometric identities and volumes of the hyperbolic twist knot conemanifolds article pdf available in journal of knot theory and its ramifications 2312 march 2014 with 55 reads. Thin position in the theory of classical knots 429 m. By the analysis of particularexamples, it is argued that, for a hyperbolic knot link, the absolute valueof this invariant grows exponentially at large n, the hyperbolic volume of the knot link complement being the growth rate.
Weeks, a computation of hyperbolic structures in knot theory, in handbook of knot theory w. Studying this hyperbolic structure can give us a lot of information about these knots. This chapter from the upcoming handbook of knot theory eds. Constructing hyperbolic polyhedra using newtons method.
Hyperbolic geometry on the figureeight knot complement. Interactive visualization of hyperbolic geometry using the. Computation of hyperbolic structures in knot theory 2008. Menasco and thistlethwaite shows how to construct hyperbolic structures on link complements and perform hyperbolic dehn filling. Physical and numerical models in knot theory series on. David futer curriculum vitae mathematics department temple university philadelphia, pa 19122 215 2047854. Etnyre knot spinning greg friedman the enumeration and classification of knots and links jim hoste knot diagrammatics louis h. This is one of many open questions that still remain in the theory of hyperbolic knots. Mostowprasad rigidity, 1968 hyperbolic structure on a. Download notes on theory of computation, this ebook has 242 pages included. Computation of hyperbolic structures in knot theory je. Geometric structures on manifolds this course will be about various kinds of geometric structures. Snappea provides for the computation of a variety of geometry invariants of the computed hyperbolic structure.
The anglesum of a triangle does not exceed two right angles, or 180. The result is that we now have better insight than ever into the structure of hyperbolic manifolds. The first part covers basic tools in hyperbolic geometry and geometric structures on 3manifolds. The latter information determines the chirality and invertibility of the link. At most nitely many hyperbolic knot complements obtained by lling one cusp of s3 lhave hidden symmetries. In the past 50 years, knot theory has become an extremely welldeveloped subject. Let l be a hyperbolic twocomponent link in s3 with crossing number at most nine. Geometries of 3manifolds by peter scott, bulletin of lms, 15 1983 online. Along with a new elementary exposition of the standard ideas from thurstons. In chapter 3, we learned that hyperbolic structures lead to developing maps and holonomy, and that the developing map is a covering map if and only if the hyperbolic stucture is complete.
This result is analogous to the computation for knots up to 15 crossings that we mentioned earlier. Workshop on volume conjecture and related topics in knot. Computation of hyperbolic structures in knot theory core. One approach to get a hyperbolic structure on a knot complement is by triangulating the knot complement.
The meaning of geometry we will use is due to klein. Hyperbolic structure on the gureeight knot complement. A survey 3 john etnyre, legendrian and transversal knots. Determining whether or not a knot is trivial is known to be in the complexity class np. The parallel postulate of euclidean geometry is replaced with. Jeff weeks, computation of hyperbolic structures in knot theory, handbook of knot theory, elsevier b. A satellite knot is the image of a nontrivial knot embedded in a solid torus k. Haraway cookie seminar, fall 2012 1 boston college. However hyperbolic geometry is difficult to visualize as many of its theorems are contradictory to similar theorems of euclidean geometry which are very familiar to us. E cient computation in wordhyperbolic groups david b. A torus knotis the image of an embedded curve on a torus under an embedding of the torus into s3. Geometric structures on the figure eight knot complement.
Thistlethwaite, editors 1 colin adams, hyperbolic knots 2 joan s. Hyperbolic geometry by cannon, floyd, kenyon, and parry. Geometric structures on 3manifolds by francis bonahon, handbook of geometric topology, available online. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined together so that it cannot be undone, the simplest knot being a ring or unknot. Neumann, zagier, volumes of hyperbolic threemanifolds, topology volume 24, issue 3, 1985, pages 307332 m. Given a hyperbolic knot, the snappea program, written by je. Acknowledgements thanks to eric schoenfeld for help with the figures.
Knot theory of complex plane curves 349 l rudolph 9. Along with a new elementary exposition of the standard ideas from thurstons work, the article includes neverbeforepublished explanations of snappeas algorithms for triangulating a link complement efficiently and for converging quickly to the. There is a very natural and easy construction of ideal triangulations and hence hyperbolic structures for oncepunctured torus bundles. Geometrical structurestypically hyperbolicprovide deep insight into the topology of knot and link complements. Arithmetic and hyperbolic structures in string theory. We can decompose the knot complement into ideal tetrahedra, giving a hyperbolic metric on the individual tetrahedra and if the gluing of these tetrahedra satis es certain conditions, then the knot complement will have a hyperbolic structure. In mathematical language, a knot is an embedding of a circle in 3dimensional euclidean space, r 3 in. Kapovich, hyperbolic manifolds and discrete groups.
Holt abstract we describe brie y some practical procedures for computing the various constants associated with a wordhyperbolic group, and report on the performance of their implementations in the kbmag package on a number of examples. Kauffman a survey of classical knot concordance charles livingston knot theory of complex plane curves lee rudolph thin position in. Adams, hyperbolic structures on knot and link complements, ph. The problem of determining the genus of a knot is known to have complexity class pspace. The group is a lie group and the space is a manifold. Knot theory is dealing with embedding of curves in manifolds of dimension 3. A knot is a single circle embedded in the affine space of dimension 3 as a smooth curve not crossing itself. This book is an introduction to hyperbolic geometry in dimension three, and its applications to knot theory and to geometric problems arising in knot theory. Along with a new elementary exposition of the standard ideas from thurstons work, the article includes neverbeforepublished explanations of snappeas algorithms for. Given a knot diagram, we color all the edges connecting the crossings by using three colors e.
It also aims to put techniques and tools from both. If we could completely understand hyperbolic structures on knot. Gutierrez mat 598 final report arizona state university, fall 2012 1 introduction and history the exact relationship between knot theory and noneuclidean geometry was a puzzle that survived more than 100 years. The histories of the two subjects were clearly intertwined. If we could completely understand hyperbolic structures on knot complements, we could completely classify hyperbolic knots. Complex hyperbolic geometry central question known results main theorem.