The three sphere famously admits a foiliation by circles: the Hopf fibration.  However, if we loosen our requirements a little bit and allow nearby circles to twist about one another, it admits infinitely many more.  Here we will construct pretty pictures of these Seifert fibrations by filling the complement of a Hopf link with torus knots.
(UNDER CONSTRUCTION)
It is natural to want to view the space of lattices in the plane as a topological space: just looking at them theres an obvious notion of when two lattices are "close".  It turns out the space of unimodular lattices is homeomorphic to the trefoil knot complement, which allows for some really pretty mathematics.  Both the modular surface and the hyperbolic plane's unit tangent bundle make surprise appearances in this story!
(UNDER CONSTRUCTION)
​Take a tetrahedron and inscribe it in a sphere.  The space of ways to do this has a natural topology on it, but what does it look like?  We can play the same game with dodecahedra, cubes, or even regular tilings of other shapes such as the Euclidean or Hyperbolic planes.  A beautiful story involving the unit tangent bundles of orbifolds can be told to help us understand these spaces.
(COMING SOON)
​Restricting ourselves to study the spaces of certain triangle tilings of the hyperbolic plane, we can construct SL(2,R)-geometric structures on a variety of manifolds.  One of these happens to be the trefoil complement, our old friend from the story of the spaces of lattices!  This connection is worked out in more detail here.
(COMING SOON)
​The modular group $\textrm{SL}(2,\mathbb{R})$ has the kernel of the "reduce mod $N$" map as a subgroup for each $N$.  In the world of orbifolds, the modular group corresponds to the $(2,3,\infty)$ triangular pillowcase, and these subgroups correspond to covers.  But what do these covers look like?  Punctured platonic solids, honeycomb patters on tori and the Klein quartic, in fact!
Life in hyperbolic 3 space is weird: for example you cant make a book or a scroll, you need to always carry a powerful flashlight with you, and if you shut your eyes its difficult to tell if you are sitting calmly on a train or laying down on a merry-go-round.  Here we'll go over the details that back up these wild claims.
We are often accustomed to viewing the three sphere through stereographic projection - but this is not what it would look like to an inhabitant.  Here, we will try to understand what it means to "look around" inside a Riemannian manifold using $\mathbb{S}^3$ as a guiding example.​
(UNDER CONSTRUCTION)
​The three strand braid group is the fundamental group of the trefoil complement - and an enlightening isomorphism can be constructed by thinking about cubic polynomials with distinct roots.  In fact this picture generalizes - there's a connection between the n strand braid group and degree n polynomials!
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