Statistics in Deformations of Large Knots
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Image Source: Incompressible surfaces in tunnel number one knot complements by Mario Eudave-Muñoz, Figure 7

Summary Project Description: A knot is an embedded circle in Euclidean 3-space. The complement of a knot in a 3-sphere is what is called a 3-manifold; a geometric object that looks locally like Euclidean 3-space. A “large” knot is one that allows for an “essential” embedded surface in its associated 3-manifold. To any manifold one can associated to it an algebraic gadget called its fundamental group (the collection of equivalent loops in the space). One can then consider what are called representations of its fundamental group; which are, ways to interpret the fundamental group as a collection of geometric motions. Understanding representations helps one under the fundamental group, and so understand the knot. A deformation of a representation is a way to “wiggle” the representations without loosing all its properties. The dimension of a deformation is how many degrees of freedom one has to “wiggle” it. This project is all about statistics of the “wiggle room” that large knots have. In particular, Dr. Lawton has determined that there exists large knots with only 1 degree of wiggle room (it is known that ALL small knots have this property). In this project, we will be gathering data on which knots in the SnapPy Census have 1 degree of freedom, and which have more. We will also 3D print (and/or VR visualize) knots and their “fundamental domains” (when they are hyperbolic). In short, we want to know how likely are "big knots in little moduli"?

Team: Dr. Lawton (professor and team lead), Mathew Hasty (PhD student and graduate leader), Savannah Crawford (undergraduate), George Andrew (undergraduate)

Introductory Movie:

Data:

  1. Computing Closed Essential Surfaces in Knot Complements (Regina Code)
  2. Motivation and first (and only) known example.
  3. MathOverFlow: Large Knots 1 and Large Knots 2

Computer Programs:

  1. SnapPy (1 hour Intro)
  2. Regina
  3. Macaulay2
  4. Computing Character Varieties (Updated Mathematica Notebook)

Knots:

  1. Knot Book
  2. Knots and Links Notes
  3. Wikipedia: Knots
  4. Rolfsen Knot Census
  5. Survey on Hyperbolic Knots
  6. Knot Invariant Note: all knots have infinite non-abelian knot groups except the unknot whose complement in the 3-sphere is a solid torus and hence has fundamental group the integers.
  7. Essential Surfaces (from Book)

Knot Visualization:

  1. TopMod Free software for topological modeling.
  2. KnotPlot Softwar for drawing knots, both in 2D and 3D
  3. SeiferView Old software, but very cool visualizations, including Knot Rollercoaster (free)

Research Papers:

  1. Incompressible surfaces in tunnel number one knot complements
  2. The SL(2,C)-character varieties of torus knots
  3. The SL(2,C) character variety of a class of torus knots
  4. Combinatorial aspects of the character variety of a family of one-relator groups
  5. On the algebraic components of the SL(2,C) character varieties of knot exteriors
  6. Varieties of Group Representations and Splittings of 3-Manifolds
  7. Plane Curves Associated to Character Varieties of 3-Manifolds (proof small knots have 1 dim moduli).
  8. Dehn Surgery on Knots (proof that hyperbolic knots have dim 1 components)
  9. https://arxiv.org/pdf/1411.5647.pdf is a paper that has satellite knots with dim greater than 1 components in the character variety.
  10. Pretzel Knots 1 and Pretzel Knots 2
  11. Morgan and Shalen:Valuations, Trees, and Degenerations of Hyperbolic Structures, I
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