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hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn
Hyperbolic_Dehn_surgery
Type of mathematical link
to hyperbolic links.[clarification needed] As a consequence of Thurston's hyperbolic Dehn surgery theorem, performing Dehn surgeries on a hyperbolic link
Hyperbolic_link
Operation used to modify three-dimensional topological spaces
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a
Dehn_surgery
Mathematical space
Consequently, there are at most three Dehn fillings of M with cyclic fundamental group. Thurston's hyperbolic Dehn surgery theorem states: M ( u 1 , u 2 , …
3-manifold
Manifold of dimension 3 equipped with a hyperbolic metric
satellite knot or a torus knot is hyperbolic. Moreover, almost all Dehn surgeries on a hyperbolic knot yield a hyperbolic manifold. A similar result is true
Hyperbolic_3-manifold
American mathematician (1946–2012)
knots and links are in fact hyperbolic. Together with his hyperbolic Dehn surgery theorem, this showed that closed hyperbolic 3-manifolds existed in great
William_Thurston
Topological complexity in mathematics
prove that hyperbolic volume decreases under hyperbolic Dehn surgery. Benedetti, Riccardo; Petronio, Carlo (1992), Lectures on hyperbolic geometry, Universitext
Simplicial_volume
Unique knot with a crossing number of four
hyperbolic knots known to have more than 6 exceptional surgeries, Dehn surgeries resulting in a non-hyperbolic 3-manifold; they have 10 and 7, respectively. A
Figure-eight knot (mathematics)
Figure-eight_knot_(mathematics)
Topics referred to by the same term
Surgery (journal), a medical journal Surgery theory, a mathematical operation used in topology; two special cases are: Dehn surgery Hyperbolic Dehn surgery
Surgery_(disambiguation)
arithmetic hyperbolic 3-manifolds with volume less than v {\displaystyle v} . This is in contrast with the fact that hyperbolic Dehn surgery can be used
Arithmetic hyperbolic 3-manifold
Arithmetic_hyperbolic_3-manifold
24 mathematical problems stated in 1982
influential 1982 paper Three-dimensional manifolds, Kleinian groups and hyperbolic geometry published in the Bulletin of the American Mathematical Society
Thurston's_24_questions
Smallest closed orientable hyperbolic 3-manifold
Fomenko–Matveev–Weeks manifold, is a closed hyperbolic 3-manifold obtained by (5, 2) and (5, 1) Dehn surgeries on the Whitehead link. It has volume approximately
Weeks_manifold
American mathematician
joint with Craig Hodgson) in exploring and clarifying Thurston's hyperbolic Dehn surgery. Kerckhoff is one of four academics from Stanford University, along
Steven_Kerckhoff
American mathematician
in low-dimensional topology whose research topics have included hyperbolic Dehn surgery and the Jones polynomial. She is a professor of mathematics at
Jessica_Purcell
Gives sufficient condition for Dehn filling to result in a negatively curved 3-manifold
condition for Dehn filling on a cusped hyperbolic 3-manifold to result in a negatively curved 3-manifold. Let M be a cusped hyperbolic 3-manifold. Disjoint
2π_theorem
theorem on sufficient conditions for Dehn surgery to produce a hyperbolic manifold,[L00] a bound on the hyperbolic volume of a knot complement of an alternating
Marc_Lackenby
distinction is behind the phenomenon of hyperbolic Dehn surgery and plays an important role in the general theory of hyperbolic 3-manifolds. William Thurston,
Algebraic_topology_(object)
3-dimensional object
1,\\0&{\text{otherwise}}.\end{cases}}\end{aligned}}} Cheerios Hyperbolic Dehn surgery Reeb foliation Whitehead manifold Doughnut Falconer, Kenneth (2004)
Solid_torus
sufficiently strong small cancellation conditions are word hyperbolic and have word problem solvable by Dehn's algorithm. Small cancellation methods are also used
Small_cancellation_theory
Normalized hyperbolic volume of the complement of a hyperbolic knot
knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume
Hyperbolic_volume
Study of mathematical knots
of topology. These topologists in the early part of the 20th century—Max Dehn, J. W. Alexander, and others—studied knots from the point of view of the
Knot_theory
in low-dimensional topology whose research topics have included hyperbolic Dehn surgery and the Jones polynomial Donald Sarason (January 26, 1933 – April
List of University of Michigan alumni
List_of_University_of_Michigan_alumni
Theorem in geometric topology
the manifold at the singularities (a process in topology called "surgery", which Max Dehn had used, already, to make Poincaré's homology sphere from a 2-sphere
Poincaré_conjecture
Three dimensional analogue of uniformization conjecture
complicated" Dehn surgeries on links, or most Haken manifolds. The geometrization conjecture implies that a closed 3-manifold is hyperbolic if and only
Geometrization_conjecture
Type of mathematical knot
four-dimensional surgery constructions. The (−2, 3, 7) pretzel knot has 7 exceptional slopes, Dehn surgery slopes which give non-hyperbolic 3-manifolds. Among
(−2,3,7)_pretzel_knot
Two interlinked loops with five structural crossings
two-cusped hyperbolic manifolds with the minimum possible volume, the other being the complement of the pretzel link with parameters (−2, 3, 8). Dehn filling
Whitehead_link
Family of quantum invariants
of framed links also give rise to invariants of 3-manifolds via the Dehn surgery construction. These invariants were discovered by Nicolai Reshetikhin
Reshetikhin–Turaev_invariant
mostly describe basic background material on hyperbolic geometry. Chapter 4 covers Dehn surgery on hyperbolic manifolds Chapter 5 covers results related
The geometry and topology of three-manifolds
The_geometry_and_topology_of_three-manifolds
finitely presented group either it is hyperbolic and satisfies a linear isoperimetric inequality or else the Dehn function is at least quadratic. The study
Van_Kampen_diagram
Topological Object
on the set H of hyperbolic 3-manifolds of finite volume. Convergence in this topology is a crucial ingredient of hyperbolic Dehn surgery, a fundamental
Geometric_topology_(object)
Techniques in topology used to produce one finite-dimensional manifold from another
manifold. s-cobordism theorem h-cobordism theorem Whitehead torsion Dehn surgery Manifold decomposition Orientation character Plumbing (mathematics) Milnor
Surgery_theory
American mathematician
Shalen proved the cyclic surgery theorem. An important corollary of the theorem is that at most one nontrivial Dehn surgery (+1 or −1) on a knot can result
Peter_Shalen
Discrete subgroup in a locally compact topological group
topology) of lattices of smaller covolume, as demonstrated by hyperbolic Dehn surgery. As lattices in rank-one p-adic groups are virtually free groups
Lattice_(discrete_subgroup)
Link formed from a finite number of twisted sections
stated about the manifolds that result from Dehn surgery on the (−2,3,7) pretzel knot in particular. The hyperbolic volume of the complement of the (−2,3,8)
Pretzel_link
polynomial. Dehn also developed Dehn surgery, which related knots to the general theory of 3-manifolds, and formulated the Dehn problems in group theory, such
History_of_knot_theory
Class of algebraic theorems
not cocompact has nontrivial deformations coming from Thurston's hyperbolic Dehn surgery theory. However, if one adds the restriction that a representation
Local_rigidity
American mathematician
109–146. Culler, Marc; Gordon, C. McA.; Luecke, J.; Shalen, Peter B. Dehn surgery on knots. Annals of Mathematics (2) 125 (1987), no. 2, 237–300. Marc
Marc_Culler
Subfield of mathematical topology
algorithm which produces a triangulated 3-manifold, given input a word (in Dehn twist generators) for the mapping class group of a surface. The 3-manifold
Computational_topology
Branch of topology
admitting a constant positively curved metric), parabolic (flat), and hyperbolic (negatively curved) according to their universal cover. The uniformization
Low-dimensional_topology
Topological manifold whose homology coincides with that of a sphere
{\displaystyle S^{3}} . Another approach is by Dehn surgery. The Poincaré homology sphere results from +1 surgery on the right-handed trefoil knot. In 2003
Homology_sphere
American mathematician
Tabachnikov) of its complete integrability. Spherical CR Geometry and Dehn Surgery, Annals of Mathematics Studies no. 165 (2007), Princeton University Press
Richard Schwartz (mathematician)
Richard_Schwartz_(mathematician)
Knot that is not equivalent to its mirror image
Listing asserted that the trefoil was chiral, and this was proven by Max Dehn in 1914. P. G. Tait found all amphichiral knots up to 10 crossings and conjectured
Chiral_knot
Hartshorne's conjectures In spherical or hyperbolic geometry, must polyhedra with the same volume and Dehn invariant be scissors-congruent? Jacobian
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Topological space that locally resembles Euclidean space
classified in the beginning of the 20th century by Poul Heegaard and Max Dehn. Poincaré pioneered the study of three-dimensional manifolds and raised a
Manifold
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HYPERBOLIC DEHN-SURGERY
HYPERBOLIC DEHN-SURGERY
HYPERBOLIC DEHN-SURGERY
HYPERBOLIC DEHN-SURGERY
HYPERBOLIC DEHN-SURGERY
HYPERBOLIC DEHN-SURGERY
HYPERBOLIC DEHN-SURGERY
HYPERBOLIC DEHN-SURGERY
HYPERBOLIC DEHN-SURGERY
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