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Page 1: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Hyperbolic Geometry and Topology

Ana Wright

September 9, 2019

Page 2: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Hyperbolic Geometry

Lobachevski (1829) and Bolyai (1832)Replacing the parallel postulate of Euclidean geometry gives usa brand new geometry!

Page 3: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Hyperbolic Geometry

Lobachevski(1829) and Bolyai(1832)Replacing the parallel postulate of Euclidean geometry gives usa brand new geometry!

Page 4: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Models for Hyperbolic Space

Upper-Half Plane

Page 5: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Models for Hyperbolic Space

Upper-Half Plane

Page 6: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Models for Hyperbolic Space

Upper-Half Plane

Page 7: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Models for Hyperbolic Space

Upper-Half Plane

Page 8: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Models for Hyperbolic Space

Upper-Half Plane Poincare Disk

Page 9: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Models for Hyperbolic Space

Upper-Half Plane Poincare Disk

Page 10: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Models for Hyperbolic Space

Upper-Half Plane Poincare Disk

Page 11: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Models for Hyperbolic Space

Upper-Half Plane Poincare Disk

Page 12: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Models for Hyperbolic Space

Upper-Half Plane Poincare Disk Beltrami-Klein

Page 13: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Models for Hyperbolic Space

Upper-Half Plane Poincare Disk Beltrami-Klein

Page 14: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Models for Hyperbolic Space

Upper-Half Plane Poincare Disk Beltrami-Klein

Page 15: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Models for Hyperbolic Space

Upper-Half Plane Poincare Disk Beltrami-Klein

Page 16: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Models for Hyperbolic Space

Upper-Half Plane Poincare Disk Beltrami-Klein

Conformal Models Nonconformal

Page 17: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Getting a Feel for Hyperbolic Space

The sum of the angles of a triangle is strictly less than π.

Page 18: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Getting a Feel for Hyperbolic Space

Ideal Triangles

Page 19: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Getting a Feel for Hyperbolic Space

AAA Congruence Theorem

Page 20: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Getting a Feel for Hyperbolic Space

AAA Congruence Theorem

Page 21: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Getting a Feel for Hyperbolic Space

AAA Congruence Theorem

Page 22: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Getting a Feel for Hyperbolic Space

AAA Congruence Theorem

Page 23: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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What about Topology?

We can’t put a hyperbolic structure on the torus...

Morally, this is because there are no hyperbolic rectangles.

Page 24: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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What about Topology?

However, we do have all-right hexagons!

Page 25: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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What about Topology?

However, we do have all-right hexagons!

Page 26: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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What about Topology?

However, we do have all-right hexagons!

Page 27: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Let’s Kick it up to 3 Dimensions!

Upper-Half Space Poincare Ball

Page 28: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Let’s Do Hyperbolic Geometry on Knots!

We can put hyperbolic structures on knot compliments inS3 (Riley, 1973)We have an explicit construction! (Thurston, 1977)Idea: Split the knot compliment into tetrahedra withvertices on the knot and glue in ideal tetrahedra fromhyperbolic space in such a way that angles work out.

Page 29: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Let’s Do Hyperbolic Geometry on Knots!

We can put hyperbolic structures on knot compliments inS3 (Riley, 1973)We have an explicit construction! (Thurston, 1977)Idea: Split the knot compliment into tetrahedra withvertices on the knot and glue in ideal tetrahedra fromhyperbolic space in such a way that angles work out.

Page 30: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Let’s Do Hyperbolic Geometry on Knots!

We can put hyperbolic structures on knot compliments inS3 (Riley, 1973)We have an explicit construction! (Thurston, 1977)Idea: Split the knot compliment into tetrahedra withvertices on the knot and glue in ideal tetrahedra fromhyperbolic space in such a way that angles work out.

Page 31: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Let’s Do Hyperbolic Geometry on Knots!

We can put hyperbolic structures on knot compliments inS3 (Riley, 1973)We have an explicit construction! (Thurston, 1977)Idea: Split the knot compliment into tetrahedra withvertices on the knot and glue in ideal tetrahedra fromhyperbolic space in such a way that angles work out.

Page 32: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Figure-Eight Knot

Page 33: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Figure-Eight Knot

Page 34: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Figure-Eight Knot

Page 35: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Figure-Eight Knot

Page 36: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Figure-Eight Knot

Page 37: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Figure-Eight Knot

Page 38: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

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Figure-Eight Knot

Page 39: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Hyperbolic Volume

The hyperbolic volume of a knot (link) which admits ahyperbolic structure is a knot (link) invariant!“Most” knots admit hyperbolic structure (hyperbolic knot).Each hyperbolic volume has a finite number of knots withthat volume.The set of hyperbolic volumes is well-ordered.It is unknown whether any hyperbolic volume is rational.It is unknown whether any hyperbolic volume is irrational!!!

Page 40: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Hyperbolic Volume

The hyperbolic volume of a knot (link) which admits ahyperbolic structure is a knot (link) invariant!“Most” knots admit hyperbolic structure (hyperbolic knot).Each hyperbolic volume has a finite number of knots withthat volume.The set of hyperbolic volumes is well-ordered.It is unknown whether any hyperbolic volume is rational.It is unknown whether any hyperbolic volume is irrational!!!

Page 41: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Hyperbolic Volume

The hyperbolic volume of a knot (link) which admits ahyperbolic structure is a knot (link) invariant!“Most” knots admit hyperbolic structure (hyperbolic knot).Each hyperbolic volume has a finite number of knots withthat volume.The set of hyperbolic volumes is well-ordered.It is unknown whether any hyperbolic volume is rational.It is unknown whether any hyperbolic volume is irrational!!!

Page 42: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Hyperbolic Volume

The hyperbolic volume of a knot (link) which admits ahyperbolic structure is a knot (link) invariant!“Most” knots admit hyperbolic structure (hyperbolic knot).Each hyperbolic volume has a finite number of knots withthat volume.The set of hyperbolic volumes is well-ordered.It is unknown whether any hyperbolic volume is rational.It is unknown whether any hyperbolic volume is irrational!!!

Page 43: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Hyperbolic Volume

The hyperbolic volume of a knot (link) which admits ahyperbolic structure is a knot (link) invariant!“Most” knots admit hyperbolic structure (hyperbolic knot).Each hyperbolic volume has a finite number of knots withthat volume.The set of hyperbolic volumes is well-ordered.It is unknown whether any hyperbolic volume is rational.It is unknown whether any hyperbolic volume is irrational!!!

Page 44: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Hyperbolic Volume

The hyperbolic volume of a knot (link) which admits ahyperbolic structure is a knot (link) invariant!“Most” knots admit hyperbolic structure (hyperbolic knot).Each hyperbolic volume has a finite number of knots withthat volume.The set of hyperbolic volumes is well-ordered.It is unknown whether any hyperbolic volume is rational.It is unknown whether any hyperbolic volume is irrational!!!

Page 45: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Hyperbolic Volume

The figure-8 knot has the smallest hyperbolic volume ofany knot: 2.02988... (Cao, Meyerhoff, 2001)The Whitehead link and the (-2,3,8) pretzel knot have thesmallest hyperbolic volume of any 2-component link:3.663862377... (Gabai, Meyerhoff, Milley, 2009)The Week’s Manifold is the unique 3-manifold with thesmallest hyperbolic volume: 0.942707... (Gabai,Meyerhoff, Milley, 2009)

Page 46: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Hyperbolic Volume

The figure-8 knot has the smallest hyperbolic volume ofany knot: 2.02988... (Cao, Meyerhoff, 2001)The Whitehead link and the (-2,3,8) pretzel knot have thesmallest hyperbolic volume of any 2-component link:3.663862377... (Gabai, Meyerhoff, Milley, 2009)The Week’s Manifold is the unique 3-manifold with thesmallest hyperbolic volume: 0.942707... (Gabai,Meyerhoff, Milley, 2009)

Page 47: Hyperbolic Geometry and Topologyawright14/Hyperbolic Geometry and Topology.pdf · The hyperbolic volume of a knot (link) which admits a hyperbolic structure is a knot (link) invariant!

university-logo-udel

Hyperbolic Volume

The figure-8 knot has the smallest hyperbolic volume ofany knot: 2.02988... (Cao, Meyerhoff, 2001)The Whitehead link and the (-2,3,8) pretzel knot have thesmallest hyperbolic volume of any 2-component link:3.663862377... (Gabai, Meyerhoff, Milley, 2009)The Week’s Manifold is the unique 3-manifold with thesmallest hyperbolic volume: 0.942707... (Gabai,Meyerhoff, Milley, 2009)