The evolution of geometric structures on 3-manifolds

McMullen, Curtis T.

doi:10.1090/s0273-0979-2011-01329-5

Cited by 17 publications

(11 citation statements)

References 46 publications

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“…(16777216 − 83886080t + 200278016t 2 − 298844160t 3 + 307822592t 4 −228524032t 5 + 123846656t 6 − 48324608t 7 + 12842496t 8 − 1930752t 9 −2544t 10 + 66288t 11 − 12587t 12 + 841t 13 )/((−3 + t)t 12 )…”

Section: Example Of the Twist Knotsmentioning

confidence: 99%

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Optimistic Limits of the Colored Jones Polynomials and the Complex Volumes Of hyperbolic Links

Cho¹

2016

J. Aust. Math. Soc.

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The optimistic limit is the mathematical formulation of the classical limit which is a physical method to expect the actual limit by using saddle point method of certain potential function. The original optimistic limit of the Kashaev invariant was formulated by Yokota, and a modified formulation was suggested by the author and others. The modified version was easier to handle and more combinatorial than the original one.On the other hand, it was known that the Kashaev invariant coincides with the evaluation of the colored Jones polynomial at the certain root of unity. The optimistic limit of the colored Jones polynomial was also formulated by the author and others, but it was so complicated and needed many unnatural assumptions.In this article, we suggest a modified optimistic limit of the colored Jones polynomial, following the idea of the modified optimistic limit of the Kashaev invariant, and show that it determines the complex volume of a hyperbolic link. Furthermore, we show that this optimistic limit coincides with the optimistic limit of the Kashaev invariant modulo 4π 2 . This new version is easier to handle and more combinatorial than the old version, and has many advantages than the modified optimistic limit of the Kashaev invariant. Because of these advantages, several applications have already appeared and more are in preparation now. * 2000 Mathematics Subject Classification: Primary 57M27; Secondly 51M25, 58J28. arXiv:1303.3701v6 [math.GT] 8 Aug 2015 1 We only consider solutions satisfying the condition that, when the potential function is expressed by W (w 1 , . . . , w n ) = ±Li 2 (w) + (extra terms), the variables inside the dilogarithms satisfy w / ∈ {0, 1, ∞}. Previously, in [20] and [7], these solutions were called essential solutions.2 The article [1] depends on this article, so using T = ∅ may look illogical. However, the proof of it in [1] relies only on Proposition 1.1 of this article and it does not require the fact T = ∅. Furthermore, all results in this article still work well if we just assume T = ∅.

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Section: Example Of the Twist Knotsmentioning

confidence: 99%

“…Some people believes it is a hint to more deeper connection between geometric and quantum topology. (See [13] for example.) After that, the generalized conjecture was proposed in [16] that 2πi lim N →∞ log L N N ≡ i(vol(L) + i cs(L)) (mod π 2 ), where cs(L) is the Chern-Simons invariant of S 3 \L defined modulo π 2 in [21].…”

Section: Introductionmentioning

confidence: 99%

Optimistic Limits of the Colored Jones Polynomials and the Complex Volumes Of hyperbolic Links

Cho¹

2016

J. Aust. Math. Soc.

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“…. measure rigidity" article by Einsiedler [Ein09], as well as the eight-page note "The mathematical work of Maryam Mirzakhani" by McMullen [McM14] and the 13-page note "The magic wand theorem of A. Eskin and M. Mirzakhani" by Zorich [Zor14,Zor]. There are a large number of surveys on translation surfaces, for example [MT02,Zor06] and the author's recent introduction aimed at a broad audience [Wri15b].…”

Section: What To Read Nextmentioning

confidence: 99%

From rational billiards to dynamics on moduli spaces

Wright

2015

Bull. Amer. Math. Soc.

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Abstract. This short expository note gives an elementary introduction to the study of dynamics on certain moduli spaces and, in particular, the recent breakthrough result of Eskin, Mirzakhani, and Mohammadi. We also discuss the context and applications of this result, and its connections to other areas of mathematics, such as algebraic geometry, Teichmüller theory, and ergodic theory on homogeneous spaces.

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“…Among the Riemannian manifolds of non-constant sectional curvature a special rôle is played by the homogeneous spaces with a large isometry group. Due to the recent approach of Hamilton-Perelman to the Poincaré conjecture (by means of Ricci flow, [8]), a great interest is in dimension 3; for a recent survey on this dimension see [12]. The present note aims to discuss two topics in 3-dimensional Riemannian geometry, both considered by Chern and Hamilton in [5]: adapted metric to a given differential 1-form and the Webster curvature W .…”

Section: Introductionmentioning

confidence: 99%

Adapted Metrics and Webster Curvature on Three Classes of 3-Dimensional Geometries

Crâşmăreanu¹

2014

International Electronic Journal of Geometry

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The Chern-Hamilton notion of adapted metric and the corresponding Webster curvature W is discussed for 3-dimensional unimodular Lie groups, Bianchi-Cartan-Vranceanu metrics and warped metrics. For the first two metrics, we control the value of W by means of two parameters: A and B provided by the Milnor frame in the former case respectively l and m in the latter case. For warped metrics, the value of W depends on the derivatives of the warping function up to order two. Bi-warped 3-metrics are introduced and studied from the point of view of Webster curvature.

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The evolution of geometric structures on 3-manifolds

Abstract: Abstract. This paper gives an overview of the geometrization conjecture and approaches to its proof.

Cited by 17 publications

References 46 publications

Optimistic Limits of the Colored Jones Polynomials and the Complex Volumes Of hyperbolic Links

Optimistic Limits of the Colored Jones Polynomials and the Complex Volumes Of hyperbolic Links

From rational billiards to dynamics on moduli spaces

Adapted Metrics and Webster Curvature on Three Classes of 3-Dimensional Geometries

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