An "orbifold Lefschetz fibration" that arises in the study of braid groups (D., Braid groups and Burnside groups)
The \Z-cover of the punctured disk and some homology classes, used to define the Burau representation (image from my dissertation).
Analyzing the boundary map in a long exact sequence for an orbifold fiber bundle. (D.-Saffat, The 3-strand braid group with torsion)
Some non-convex regular polyhedra (see Coxeter, Regular Complex Polytopes)
A braid trivial under Burau is trivializable with certain relations. (D. The Burau Rep. and Shapes of Polyhedra)
Seeking generators for homology of an abelian cover of the punctured disk.
MCG of #^n (S^2 x S^1) yielding the Nielsen generators of Out(F_n). (see Brendle, Broaddus, Putman)
The trefoil knot complement deformation retracts to a 2-complex.
The figure eight knot is fibered because it is a Murasugi sum of Hopf bands. (see Baader and Graf "cord criterion")
A cartoon about how to construct arithmetic lattices. (see Witte Morris, Introduction to Arithmetic Groups)
A Brunnian braid, i.e. a braid which becomes trivial if you delete any one of its strands.
The moduli space of equi-angular hexagons is a convex polyhedron in hyperbolic space. (see Thurston, Shapes of Polyhedra)
Wallpaper groups whose points groups are Z/2Z. (see Hiller, Wallpaper groups and group cohomology)
Working out an example about braid foliations. (see Birman, Hirsch, Unknot recognition algorithm)
A nice complex-valued function on a cyclic branched cover of the disk. (see McMullen, Braid Groups and Hodge Theory)
Understanding how the mapping class group acts on the space of polyhedra. (see Thurston, Shapes of Polyhedra)
An old pet project: what is the longest line segment that you can "turn around" on a given curve? https://www.desmos.com/calculator/9fb4e229