a journey through hyperbolic spaceawright14/a journey through... · a journey through hyperbolic...
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![Page 1: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/1.jpg)
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A Journey Through Hyperbolic Space
Ana Wright
April 16, 2020
![Page 2: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/2.jpg)
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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: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/3.jpg)
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Hyperbolic Geometry
Lobachevski(1829) and Bolyai(1832)Replacing the parallel postulate of Euclidean geometry gives usa brand new geometry!
![Page 4: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/4.jpg)
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Hyperbolic Space
Hyperbolic geometry is geometry on a surface with constantnegative Gaussian curvature.
![Page 5: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/5.jpg)
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Hyperbolic Space
Hyperbolic geometry is geometry on a surface with constantnegative Gaussian curvature.
![Page 6: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/6.jpg)
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Hyperbolic Space
What are “straight lines” on a curved surface?Geodesics!
![Page 7: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/7.jpg)
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Hyperbolic Space
Parallel postulate
![Page 8: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/8.jpg)
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Hyperbolic Space
Parallel postulate
![Page 9: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/9.jpg)
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Models for Hyperbolic Space
Upper-Half Plane
![Page 10: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/10.jpg)
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Models for Hyperbolic Space
Upper-Half Plane
![Page 11: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/11.jpg)
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Models for Hyperbolic Space
Upper-Half Plane
![Page 12: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/12.jpg)
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Models for Hyperbolic Space
Upper-Half Plane
![Page 13: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/13.jpg)
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Models for Hyperbolic Space
Upper-Half Plane Poincare Disk
![Page 14: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/14.jpg)
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Models for Hyperbolic Space
Upper-Half Plane Poincare Disk
![Page 15: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/15.jpg)
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Models for Hyperbolic Space
Upper-Half Plane Poincare Disk
![Page 16: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/16.jpg)
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Models for Hyperbolic Space
Upper-Half Plane Poincare Disk
![Page 17: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/17.jpg)
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Models for Hyperbolic Space
Upper-Half Plane Poincare Disk Beltrami-Klein
![Page 18: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/18.jpg)
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Models for Hyperbolic Space
Upper-Half Plane Poincare Disk Beltrami-Klein
![Page 19: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/19.jpg)
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Models for Hyperbolic Space
Upper-Half Plane Poincare Disk Beltrami-Klein
![Page 20: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/20.jpg)
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Models for Hyperbolic Space
Upper-Half Plane Poincare Disk Beltrami-Klein
![Page 21: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/21.jpg)
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Models for Hyperbolic Space
Upper-Half Plane Poincare Disk Beltrami-Klein
Conformal Models Nonconformal
![Page 22: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/22.jpg)
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Getting a Feel for Hyperbolic Space
The sum of the angles of a triangle is strictly less than π.
![Page 23: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/23.jpg)
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Getting a Feel for Hyperbolic Space
The sum of the angles of a triangle is strictly less than π.
![Page 24: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/24.jpg)
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Getting a Feel for Hyperbolic Space
Ideal Triangles
![Page 25: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/25.jpg)
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Getting a Feel for Hyperbolic Space
AAA Congruence Theorem
![Page 26: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/26.jpg)
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Getting a Feel for Hyperbolic Space
AAA Congruence Theorem
![Page 27: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/27.jpg)
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Getting a Feel for Hyperbolic Space
AAA Congruence Theorem
![Page 28: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/28.jpg)
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Getting a Feel for Hyperbolic Space
AAA Congruence Theorem
![Page 29: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/29.jpg)
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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 30: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/30.jpg)
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What about Topology?
However, we do have all-right hexagons!
![Page 31: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/31.jpg)
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What about Topology?
However, we do have all-right hexagons!
![Page 32: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/32.jpg)
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What about Topology?
However, we do have all-right hexagons!
![Page 33: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/33.jpg)
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Let’s Kick it up to 3 Dimensions!
Upper-Half Space Poincare Ball
![Page 34: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/34.jpg)
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Let’s Do Hyperbolic Geometry on Knots!
A knot is a circle which is “knotted” in 3-dimensional space.
We are going to put our knots in a 3-sphere (S3).
![Page 35: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/35.jpg)
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Let’s Do Hyperbolic Geometry on Knots!
A knot is a circle which is “knotted” in 3-dimensional space.
We are going to put our knots in a 3-sphere (S3).
![Page 36: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/36.jpg)
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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 37: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/37.jpg)
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 38: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/38.jpg)
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 39: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/39.jpg)
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 40: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/40.jpg)
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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.It is unknown whether any hyperbolic volume is rational.It is unknown whether any hyperbolic volume is irrational!!!
![Page 41: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/41.jpg)
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.It is unknown whether any hyperbolic volume is rational.It is unknown whether any hyperbolic volume is irrational!!!
![Page 42: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/42.jpg)
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.It is unknown whether any hyperbolic volume is rational.It is unknown whether any hyperbolic volume is irrational!!!
![Page 43: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/43.jpg)
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.It is unknown whether any hyperbolic volume is rational.It is unknown whether any hyperbolic volume is irrational!!!
![Page 44: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/44.jpg)
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.It is unknown whether any hyperbolic volume is rational.It is unknown whether any hyperbolic volume is irrational!!!
![Page 45: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/45.jpg)
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Figure-Eight Knot
![Page 46: A Journey Through Hyperbolic Spaceawright14/A Journey Through... · A Journey Through Hyperbolic Space Ana Wright April 16, 2020. university-logo-udel Hyperbolic Geometry ... Split](https://reader036.vdocuments.mx/reader036/viewer/2022070111/604ed6f76da02634f013064c/html5/thumbnails/46.jpg)
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Figure-Eight Knot