Thrice-punctured spheres in hyperbolic $3$-manifolds

Adams, Colin

doi:10.1090/s0002-9947-1985-0768730-6

Cited by 53 publications

(132 citation statements)

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“…Repeating the process depicted there, and then cutting open along a single twicepunctured disk and twisting a half-twist before regluing yields the link depicted in Figure 2. D FIGURE 2 w, FIGURE 3 Corollaries 5.1 and 5.2 of [1] imply that the result is hyperbolic with volume 2(772 -1) times the volume of the Whitehead link, where the volume of the Whitehead link is given approximately by 3.6638.…”

Section: Ball In Mo Given By N(d)un(g)mentioning

confidence: 95%

“…This will be the contents of §2. The results listed above will then follow from results in [1] and [2].…”

Section: Introductionmentioning

confidence: 93%

“…However, since each of φ\ and Φ2 lifts to parabolic isometries of H 3 ', we have the parabolic isometry corresponding to μ that is the product of the two parabolic isometries corresponding to φ\ and φ^. In particular, this implies that φ\ and Φ2 generate a Fuchsian subgroup of the fundamental group, that Fuchsian subgroup corresponding to a totally geodesic possibly immersed thrice-punctured sphere (see [1]). …”

Section: M-n(μ)mentioning

confidence: 99%

“…Let M3 be a manifold obtained by gluing these two manifolds together along Pi and Dι and also along P2 and Z?2 The original manifold Mo can be obtained from Dehn filling the resulting manifold M3. FIGURE 4 However, by cutting open along twice-punctured disks in Ms corresponding to the twice-punctured disks in S 3 -L n , twisting a halftwist and then regluing, we can obtain numerous manifolds, all of which are hyperbolic and all of which have the same volume by Theorem 4.1 of [1]. However, they cannot be homeomorphic as they have different numbers of cusps.…”

Section: Theorem 11 (A) Let V O Be Any Positive Real Number Then Tmentioning

confidence: 99%

“…Alexander [3] proved that compact orientable surface bundles over S 1 with torus boundary components form a complete family of ancestor manifolds. Myers [15] and Gonzalez-Acuna [7] showed that in fact compact orientable surface bundles over S 1 with a single torus boundary component form a complete family of parent manifolds.…”

Section: Introductionmentioning

confidence: 99%

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Dehn filling hyperbolic 3-manifolds

Adams¹

1994

Pacific J. Math.

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Section: Ball In Mo Given By N(d)un(g)mentioning

confidence: 95%

“…This will be the contents of §2. The results listed above will then follow from results in [1] and [2].…”

Section: Introductionmentioning

confidence: 93%

Section: M-n(μ)mentioning

confidence: 99%

Section: Theorem 11 (A) Let V O Be Any Positive Real Number Then Tmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dehn filling hyperbolic 3-manifolds

Adams¹

1994

Pacific J. Math.

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Guts of Surfaces and the Colored Jones Polynomial

Futer

Kalfagianni

Purcell

2013

Lecture Notes in Mathematics

198

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PrefaceAround 1980, W. Thurston proved that every knot complement satisfies the geometrization conjecture: it decomposes into pieces that admit locally homogeneous geometric structures. In addition, he proved that the complement of any non-torus, non-satellite knot admits a complete hyperbolic metric which, by the Mostow-Prasad rigidity theorem, is is necessarily unique up to isometry. As a result, geometric information about a knot complement, such as its volume, gives topological invariants of the knot.Since the mid-1980's, knot theory has also been invigorated by ideas from quantum physics, which have led to powerful and subtle knot invariants, including the Jones polynomial and its relatives, the colored Jones polynomials. Topological quantum field theory predicts that these quantum invariants are very closely connected to geometric structures on knot complements, and particularly to hyperbolic geometry. The volume conjecture of R. Kashaev, H. Murakami, and J. Murakami, which asserts that the volume of a hyperbolic knot is determined by certain asymptotics of colored Jones polynomials, fits into the context of these predictions. Despite compelling experimental evidence, these conjectures are currently verified for only a few examples of hyperbolic knots.This monograph initiates a systematic study of relations between quantum and geometric knot invariants. Under mild diagrammatic hypotheses that arise naturally in the study of knot polynomial invariants (A-or B-adequacy), we derive direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. We prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement, and that certain coefficients of the polynomial measure how far this surface is from being a fiber in the knot complement. In particular, the surface is a fiber if and only if a certain coefficient vanishes.Our results also yield concrete relations between hyperbolic geometry and colored Jones polynomials: for certain families of links, coefficients of the polynomials determine the hyperbolic volume to within a factor of 4. Our methods here provide a deeper and more intrinsic explanation for similar connections that have been previously observed.Our approach is to generalize the checkerboard decompositions of alternating knots and links. For A-or B-adequate diagrams, we show that the checkerboard knot surfaces are incompressible, and obtain an ideal polyhedral decomposition of their complement. We use normal surface theory to establish a dictionary between the pieces of the JSJ decomposition of the surface complement and the combinatorial structure of certain spines of the checkerboard surface (state graphs). In particular, we give a combinatorial formula 4 for the complexity of the hyperbolic part of the JSJ decomposition (the guts) of the surface complement in terms of the diagram of the knot, and use this to give lower bounds on volumes of several ...

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An algorithm producing hyperbolicity equations for a link complement in S 3

Petronio

1992

Geom Dedicata

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We describe a constructive and effective method for decomposing the complement of an alternating link in the three-sphere into tetrahedra with identifications and vertices removed. Consequently we obtain an algorithm for writing down the byperbolicity equations associated to such decomposition.According to the combined results of [Thl], ] and [Ep.-Pe] (see also [Ne-Za] and [Be-Pc, Chapter El, in the latter of which one could find a selfcontained exposition of the subject) an orientable non-compact hyperbolic three-manifold of finite volume can be decomposed in a finite number of tetrahedra with glued faces and vertices removed; these tetrahedra are isometric to ideal tetrahedra in the hyperbolic three-space and they are parametrized by complex numbers satisfying certain rational equations depending on the combinatorics of the gluings (see Section 3 for definitions and precise statements). Moreover it is well known that such a manifold is the interior of a compact manifold whose boundary consists of tori. The natural question, arising from this topological description, whether the complement of a link in S a can be endowed with a hyperbolic structure, was faced and given important answers in [Th3], [Adl], [Ad2] and [Me2], the results being purely existential as based on Thurston's hyperbolization theorem ([Th2] and [Th4]). On the other hand one can produce examples using the fact that for any three-manifold having a topological decomposition into tetrahedra as the one sketched above the equations can be written down and a solution, if any, actually defines a hyperbolic structure on the manifold.As suggested by these remarks in this paper we provide an algorithm for writing down hyperbolicity equations (associated to a decoraposition into tetrahedra) for the complement of a link in S 3 (not an arbitrary one, but quite a general one; moreover we prove that it is always possible to add components to a link in order to get one to which the algorithm applies).Some basic examples are studied in [Thl] (Thurston's notes are actually the main source for this subject); in [Mel] a description of a more general procedure is partially carried out but in a quite rough way (for instance the conditions under which the procedure works are not explicitly stated; on the contrary we collect those we need in step 1 of our algorithm). We actually Geometriae Dedicata 44: 67-104, 1992.

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Thrice-punctured spheres in hyperbolic $3$-manifolds

Cited by 53 publications

References 5 publications

Dehn filling hyperbolic 3-manifolds

Dehn filling hyperbolic 3-manifolds

Guts of Surfaces and the Colored Jones Polynomial

An algorithm producing hyperbolicity equations for a link complement in S 3

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