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127
Twisted Alexander polynomials detect fibered 3-manifolds
- Monopoles and Three-Manifolds, New Mathematical Monographs (No. 10), Cambridge University Press. , Knots, sutures and excision
"... Abstract. A classical result in knot theory says that for a fibered knot the Alexander polynomial is monic and that the degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3–manifolds. In this paper we show that ..."
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Cited by 40 (11 self)
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Abstract. A classical result in knot theory says that for a fibered knot the Alexander polynomial is monic and that the degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3–manifolds. In this paper we show that the conditions on twisted Alexander polynomials are not only necessary but also sufficient for a 3–manifold to be fibered. By previous work of the authors this result implies that if a manifold of the form S 1 × N 3 admits a symplectic structure, then N fibers over S 1. In fact we will completely determine the symplectic cone of S 1 × N in terms of the fibered faces of the Thurston norm ball of N. 1.
Gravitational Lensing from a Spacetime Perspective
, 2004
"... The theory of gravitational lensing is reviewed from a spacetime perspective, without quasi-Newtonian approximations. More precisely, the review covers all aspects of gravita-tional lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature. It i ..."
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Cited by 27 (3 self)
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The theory of gravitational lensing is reviewed from a spacetime perspective, without quasi-Newtonian approximations. More precisely, the review covers all aspects of gravita-tional lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature. It includes the basic equations and the relevant techniques for calcu-lating the position, the shape, and the brightness of images in an arbitrary general-relativistic spacetime. It also includes general theorems on the classification of caustics, on criteria for multiple imaging, and on the possible number of images. The general results are illustrated with examples of spacetimes where the lensing features can be explicitly calculated, including the Schwarzschild spacetime, the Kerr spacetime, the spacetime of a straight string, plane gravitational waves, and others.
Rotational symmetry of self-similar solutions to the Ricci flow
- Invent. Math
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Nonholonomic Ricci flows. II. Evolution equations and dynamics
- J. Math. Phys
"... This is the second paper in a series of works devoted to nonholonomic Ricci flows. Following our idea that imposing non–integrable (nonholonomic) constraints on Ricci flows of Riemannian metrics, we model mutual transforms of generalized Finsler–Lagrange and Riemann geometries. There are verified so ..."
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Cited by 19 (18 self)
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This is the second paper in a series of works devoted to nonholonomic Ricci flows. Following our idea that imposing non–integrable (nonholonomic) constraints on Ricci flows of Riemannian metrics, we model mutual transforms of generalized Finsler–Lagrange and Riemann geometries. There are verified some assertions made in the first partner paper and developed a formal scheme in which the geometric constructions are elaborated for the canonical nonlinear and linear connections. The scheme is applied to a study of Hamilton’s Ricci flows on nonholonomic manifolds and related Einstein spaces and Ricci solitons. The nonholonomic evolution equations are derived from Perelman’s functionals redefined in a form to be adapted to the nonlinear connection structure. Finally, a statistical analogy for nonholonomic Ricci flows is formulated and the corresponding thermodynamical values are computed for compact configurations.
The minimal volume orientable hyperbolic 2-cusped 3-manifolds
, 2010
"... We prove that the Whitehead link complement and the (−2, 3, 8) pretzel link complement are the minimal volume orientable hyperbolic 3-manifolds with two cusps, with volume 3.66... =4 × Catalan’s constant. We use topological arguments to establish the existence of an essential surface which provides ..."
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Cited by 18 (0 self)
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We prove that the Whitehead link complement and the (−2, 3, 8) pretzel link complement are the minimal volume orientable hyperbolic 3-manifolds with two cusps, with volume 3.66... =4 × Catalan’s constant. We use topological arguments to establish the existence of an essential surface which provides a lower bound on volume and strong constraints on the manifolds that realize that lower bound.
Guts of surfaces and the colored Jones polynomial
"... This work 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 ..."
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Cited by 17 (5 self)
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This work 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 particular 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.
MINIMALLY INVASIVE SURGERY FOR RICCI FLOW SINGULARITIES
"... Abstract. In this paper, we construct smooth forward Ricci flow evolutions of singular initial metrics resulting from rotationally symmetric neckpinches on S n+1, without performing an intervening surgery. In the restrictive context of rotational symmetry, this construction gives evidence in favor o ..."
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Cited by 15 (2 self)
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Abstract. In this paper, we construct smooth forward Ricci flow evolutions of singular initial metrics resulting from rotationally symmetric neckpinches on S n+1, without performing an intervening surgery. In the restrictive context of rotational symmetry, this construction gives evidence in favor of Perelman’s hope for a “canonically defined Ricci flow through singularities”. Contents
Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3-manifolds
, 2008
"... We apply an equivariant version of Perelman’s Ricci flow with surgery to study smooth actions by finite groups on closed 3-manifolds. Our main result is that such actions on elliptic and hyperbolic 3-manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott ..."
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Cited by 15 (0 self)
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We apply an equivariant version of Perelman’s Ricci flow with surgery to study smooth actions by finite groups on closed 3-manifolds. Our main result is that such actions on elliptic and hyperbolic 3-manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott [MS86], it follows that such actions on geometric 3-manifolds (in the sense of Thurston) are always geometric, i.e. there exist invariant locally homogeneous Riemannian metrics. This answers a question posed by Thurston in [Th82].
