VOLUMES OF TWO-BRIDGE CONE MANIFOLDS IN SPACES OF CONSTANT CURVATURE
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Springer Science+Business Media New York (2020)
VOLUMES OF TWO-BRIDGE CONE MANIFOLDS IN SPACES OF CONSTANT CURVATURE A. D. MEDNYKH∗ Sobolev Institute of Mathematics pr. Koptuga, 4 630090, Novosibirsk, Russia Novosibirsk State University Pirogova st., 2 630090, Novosibirsk, Russia [email protected]
Devoted to the memory of Ernest Borisovich Vinberg who was a true Man, Teacher and Mathematician
Abstract. We investigate the existence of hyperbolic, spherical or Euclidean structure on cone-manifolds whose underlying space is the three-dimensional sphere and singular set is a given two-bridge knot. For two-bridge knots with not more than 7 crossings we present trigonometrical identities involving the lengths of singular geodesics and cone angles of such cone-manifolds. Then these identities are used to produce exact integral formulae for the volume of the corresponding cone-manifold modeled in the hyperbolic, spherical and Euclidean geometries.
Contents 1. 2. 3. 4.
Introduction Preliminaries Trefoil knot and other toric knots The Figure eight knot 41 4.1. Cone–manifold 41 (α) and its properties 4.2. Hyperbolic volume of 41 (α) 4.3. Spherical volume of 41 (α) 4.4. Specific Euclidean volume of 41 (α) 5. The three–twist knot 52 5.1. Hyperbolic volume of 52 (α) 5.2. Spherical volume of 52 (α) 5.3. Specific Euclidean volume of 52 (α) DOI: 10.1007/S00031-020-09632-x The research was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (Contract No. 14.Y26.31.0025). Received June 26, 2020. Accepted September 19, 2020. Corresponding Author: A. D. Mednykh, e-mail: [email protected] ∗
A. D. MEDNYKH
6. 7. 8. 9. 10. 11. 12. 13.
Stevedore's Knot 61 Knot 62 Knot 63 Knot 72 Knot 73 Knot 74 Knots 75 , 76 and 77 Tables 1. Introduction
In 1975 R. Riley found examples of hyperbolic structures on some knot and link complements in the three-dimensional sphere. Seven of them, so called excellent knots, were described in [R2]. One more, very important case of the figure eight knot was investigated in his manuscript [R3] (see also commentary on Robert Riley's results in [BJS]). Later, in the spring of 1977, W.P. Thurston announced an existence theorem for Riemannian metrics of constant negative curvature on three-dimensional manifolds. In particular, it turned out that the knot complement of a simple knot (excepting torical and satellite) admits a complete hyperbolic structure. This fact allowed us to consider knot theory as a part of the geometry and theory of discrete groups. Starting from Alexander’s work [A], polynomial invariants became a convenient instrument for study of knots and links. A lot of different kinds of such polynomials were discovered in the last two decades of the 20th century. Among them we mention the Jones, Kaufmann bracket and HOMFLY-PT polynomials, complex distance polynomial, A-polynomial and others; see [K], [CCGLS], [MR], [DMM], [HLM2]. This relates the knot theory with algebra and algebraic geometry. The algebraic technique is used to find the most important geometric
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