Knot Page

Knot Links and Hyperbolic 3-Manifolds

History

Scottish Physics and Knot Theory's Odd Origins PDF - D. S. Silver, or here and here
Birth of P G Tait, Pioneer of Knot Theory - APS News
Orbits of Asteroids, a Braid, and the First Link Invariant - M. Epple
The modern legacies of Thomson's atomic vortex theory in classical electrodynamics - S. J. Lomonaco
Life and Scientific Work of Peter Guthrie Tait - book, online
History of Knot Theory - J. H. Przytycki
Kelvin's Brilliant Vortex Atoms
A Brief History of Knot Theory - E. Colberg
Knot theory from Vandermonde to Jones
Tait's sealed envelope

Sites

Knot Atlas
The Geometry Junkyard - more links!
Parametrized Knots
Polynomial knots
SeifertView
Seifert Matrix Computations - J. Collins
Calabi-Yau and Hanson’s surfaces
KnotFinder at https://knotinfo.math.indiana.edu/index2.php
SnapPy - topology and geometry of 3-manifolds
Regina - this also has triangulations
Archive of papers and biographies

Videos

Peter Guthrie Tait: A Knot's Tale - J. Collins
Virtual Knot Theory - L. Kauffmann
Lectures at Andrews University - A. Bosman
Talks on knot theory hosted by R. Fenn and L. Kauffman
Is the Conway knot slice? (After Lisa Piccirillo)
How You Too Can Solve 50+ Year Old Problems - Lisa Piccirillo
Knots and Surfaces - Wildberger lecture
Knots, three-manifolds and instantons
Landau Lectures| Prof. Thurston | Part 1 | 1995/6, Part 2
The film Not Knot and article
- In case you missed Flatland
  - Flatland: A Romance of Many Dimensions
    - html book
    - Sequels and other sources like Sphereland
    - TED-Ed 5 minute story

The evolution of geometric structures on 3-manifolds

Structure of hyperbolic manifolds - Lecture 1

Segerman
- Visualizing hyperbolic honeycombs (31:42)

New Links (Posted June 2022)

Untangling Why Knots Are Important - S. Strogatz interview with C. Adams and L. Piccirillo
Is the Conway knot slice? (23:59)

Graphics

PLink - Python tool, also part of SnapPy
- Type Manifold('4_1').plink() to edit knot 4_1
- or, type
  - M = Manifold('4_1')
  - M.volume() - computes volume
  - M.plink() - To go to Link Editor
  - M.browse() - Gives added info; go inside!
- Screenshots
Topology Software
SnapPy Github site - 'interior views of a hyperbolic 3-manifold'
7-minute tutorial
Seifert surface for (5,3) torus knot
TpX: Drawing Tool for LaTeX [Save to as TpX and edit]
Drawing hyperbolic polyhedra

Figure 8 Knot Complement

Figure Eight Knot and at Knot Atlas
Seifert Surface
A Topological Picturebook - Figure Eight
Complement of Figure-8 knot
Figure 8 Knot Complement Model and video, Segerman's page
- Visualizing hyperbolic honeycombs - Segerman
Sculptures in S^3 - in The Bridges Archive, shadertoy rendition from here.
Hopf Fibration - map from a hypersphere in 4D onto a sphere in 3D, Johnson lecture
Volumes of Hyperbolic Link Complements - Good discussion of the topology and geometry
William Thurston
- Landau Lectures| Prof. Thurston | Part 1, Part 2
- Knots to Narnia
- The Mystery of 3-Manifolds (58:40)
  - Borromean rings and cubes is related to talk (1:12)
  - Impossible Triangle Possible in Nil Geometry (2:08)
- Geometry and topology of three-manifolds - lecture notes
What's Next? The mathematical legacy of Bill Thurston, Videos from 2014 conference
The Shape of Space - Amazon, Library Catalog
- Jeff Weeks Lecture
- Geometry Center Video
- Exploring the Shape of Space pdf
Visualizing Regular Tessellations (1:01:27)
Practical computation with hyperbolic 3 manifolds - Dunfield (1:06:59)
Hyperbolic Knot Theory - Lecture 1 (1:07:27) , Lecture 2 (1:13:54)
Cusps of Hyperbolic Knots - slides
The geometry of knot complements - slides
Hyperbolic Knot Theory - thesis
Hyperbolic Geometry on the Figure-Eight Knot Complement - report
Hyperbolic Structure of Knot Complements
The figure-eight knot complement Geometric structures on manifolds
Tetrahedral hyperbolic 3-manifolds and links - Talk
Hyperbolic Non-Euclidean World

Hyperbolic 3-Manifolds

What is a Hyperbolic 3-Manifold? - C. Adams (2018)
The geometry of 3-manifolds 1/4 (1:00:53)
Hyperbolic Geometry
- Hyperbolic Geometry: The first 150 Years - J. Milnor
- Notes from Warwick - 130 pgs
- Poincare' Upper Half Plane Model and Poincare' Disk model - Royster
- Area of hyperbolic triangle - Example
- A Roundabout Introduction to Hyperbolic Area
- Drawing Graphs in the Hyperbolic Plane
- Conformal Models of the Hyperbolic Geometry
- Hyperbolic Geometry in Geometry with an Introduction to Cosmic Topology
- Hyperbolic geometry, surfaces, and 3-manifolds - See Chapter 2
Areas and Volumes
- Heron's Formula and Brahmagupta's Generalization
- Piero della Francesca's Tetrahedron Formula
- The Geometry of Piero della Francesco
- Tartaglia's Formula - General Trattato di Numeri et Misure, Part IV, Book 2, p. 35
- Simplex Volumes and the Cayley-Menger Determinant
- Hyperbolic Volumes and Symmetry
- Brahmahupta’s theorem - generalization - Mednykh Slides
Hyperbolic Volumes
- Computation of Volume J. W. Milnor in Thurston's Notes
- Formulas on hyperbolic volume - general
- The many aspects of hyperbolic volume (1:02:18) - Kellerhals
- Schläfli – for lush, voluminous polyhedra - blog
- On Application Of Contemporary Proof Of The Sforza Formula To Computation Of Volumes Of Hyperbolic Tetrahedra Of Special Kind
- Explicit Volume Formulafor A Hyperbolic Tetrahedronin Terms Of Edge Lengths
- Volumes of polytopes in spaces of constant curvature
- Hyperbolic Volumes and Symmetry
- The Seidel problem - Abrosimov
- Abhijit Champanerkar: Hyperbolic Knot Theory, Lecture-II (1:08:50)
The Dilogarithm Function
Hyperbolic Knot Theory - Purcell
Tetrahedral hyperbolic 3-manifolds and links - Slides and Lecture (1:11:01)

Papers

Miscellaneous

Graduate Student Solves Decades-Old Conway Knot Problem
Knot Theory from Cornell's Math Explorer's Club
Visualization of Seifert surface https://www.maths.ed.ac.uk/~v1ranick/papers/vanwijk.pdf
Visualization of the Genus of Knots https://www.win.tue.nl/~vanwijk/knot.pdf
Seifert Matrix Julia Collins https://www.maths.ed.ac.uk/~v1ranick/julia/SeifertMatrix.pdf or https://pdfs.semanticscholar.org/4519/393e6c288a5248ec21619e673db886e0e057.pdf
Visualizing Torus Knots - play with sliders
Seifert Surface Creator - in Python
Computing The Alexander Polynomial From Seifert Surfaces
Baez Braid Group notes
Geometry of Knot Complements
Class on 3-Manifolds
Hyperbolic Knot Theory - Purcell, Lecture
Tiling Hyperbolic Space
Alice, Bob, and the average shadow of a cube - video
Visualizing the sphere and the hyperbolic plane
Visualizing hyperbolic honeycombs - Segerman
Curved Spaces simulator - Weeks
hypVR-Ray - simulation

Background Math

What is a Manifold - Topological Manifolds (See lessons 6-7)
Lie Groups and Lie Algebras

Books

An Introduction to Geometric Topology - Martelli
Hyperbolic Knot Theory - Purcell
Geometry with an Introduction to Cosmic Topology - Hitchman
Geometry and topology of three-manifolds - Thurston
The Shape of Space, 3rd Ed. - Jeffrey R. Weeks

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does not necessarily represent UNC Wilmington official information.