(More explanations to come)

My Papers




14. (with Christian Millichap and Will Worden) Symmetries and hidden symmetries of \((\epsilon,d_L)\)-twisted knot complements arxiv
13.(with Robert Haraway) On the complexity of cusped non-hyperbolicity arxiv
12. (with Ken Baker and Joan Licata) Jointly primitive knots and surgeries between lens spaces. arxiv
11. (appendix to Ken Baker's paper) The Poincare homology sphere, lens space surgeries, and some knots with tunnel number two arxiv.
10. (with Nathan Sunukjian) Surfaces in 4-manifolds: smooth isotopy arxiv<



9. (with Jessica Purcell) Geometry of planar surfaces and exceptional fillings

We exhibit a family of hyperbolic 3-manifolds that each have a reducible filling along a multislope such that each component has length \(\frac{10}{\sqrt{3}}-\epsilon\) (with \( \epsilon\) arbitrarily small of course). We also show that any 2D horoball packing of a surface can be obtained as the induced horoball packing on an embedded surface in a 3-manifold. We conjecture that for all 2D horoball packings on punctured spheres there is at least one cusp of lenght below \(\frac{10}{\sqrt{3}}-\epsilon\).


8. (with Stavros Garoufalidis, Craig Hodgson, and Hyam Rubinstein) The 3D-index and normal surfaces We show that 3-manifold invariant the 3D index can be expressed as a sum over solutions to the normal surface equations which have at most 2 quad types per tetrahedra.

7. (with Genevieve Walsh) The Big Dehn surgery graph and the link of $S^3.$ arxiv

This paper describes a graph that was known to the experts, but never appeared in the literature. This graph has a vertex for each closed, orientable 3-manifold and an edge between two vertices if the corresponding manifolds can be related by Dehn surgery along ONE simple closed curve embedded in one of the manifolds. We call this graph the Big Dehn surgery graph.

We then show several properties of the graph including: the graph is connected, the graph has infinte diameter, the graph is not delta-hyperbolic. Furthermore, we explore the relations between properties of the fundamental group of a manifold and if that manifold can be obtained from surgery along a knot in $S^3$. The weight of the group is of particular interest here. A group $G$ is weight n if there exists a set of $n$ conjugacy classes in $G$ that generate the group and no set of $n-1$ conjugacy classes generate $G$. (By convention, the trivial group is weight 1.) We also show that there are hyperbolic manifolds with weight 1 fundamental groups that are not surgery along a knot in $S^3$. The paper also asks a number of interesting questions about this graph.


6. (with Nathan Dunfield and Joan Licata) Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling

We find examples of 1-cusped asymmetric manifolds with two lens space fillings and then use those to construct hyperbolic L-spaces which are not the double branched cover of a knot or link.


5. (with Kazuhiro Ichihara, Masahide Kashiwagi, Hidetoshi Masai, Shin'ichi Oishi, and Akitoshi Takayasu) Verified computations for hyperbolic 3-manifolds submitted, arxiv code
This work provides a rigorous verfication of the numerical approximation of SnapPea or SnapPy. Although Snap has similar abilities for small complexity manifolds, our methods seem to be applicable to a strictly larger set of manifolds. These methods were the crux of large scale implementations by Ichihara and Masai and Martelli, Petronio, and Roukema.

We also show that all cusped orientable census manifolds are hyperbolic and all closed orientable census manifolds are hyperbolic. The later required producing a set of triangulations that were able to verified. In particular, the triangulations in question had to have the property that all tetrahedral parameters in a solution to the gluing equations had positive imaginary part. The triangulations we used along with our code which runs as python module is available here http://www.oishi.info.waseda.ac.jp/~takayasu/hikmot/.

4. (with Ken Baker and Brandy Guntel) On manifolds with multiple cyclic fillings, and generalized Berge knots. submitted, arxiv


3. On knot complements that decompose into regular ideal dodecahedra, to appear in Geometriae Dedicata, arxiv The computation written as a magma file is here.


2. Small knot complements, exceptional surgeries and hidden symmetries, submitted arxiv


1. Commensurability classes containing three knot complements, Alg. & Geo. Top, Vol 10 (2010) pdf