New paper: “Distributions of distances and volumes of balls in homogeneous lens spaces” with Chris Peterson and our student Brenden Balch

New paper with Jason Cantarella, Tetsuo Deguchi, and Erica Uehara, in which we continue our quest to turn topological polymer problems into spectral graph theory problems (really, linear algebra problems):

Vanishing Point

Conformal image of the square grid in the infinite strip \(\{z \in \mathbb{C} : 1 \leq \operatorname{Re}(z) \leq 2\}\) under the map \(z \mapsto \frac{-4i}{z}\).

Source code:

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Minimal-stick examples of the knots \(9_{35}\), \(9_{39}\), \(9_{43}\), \(9_{45}\), and \(9_{48}\).

Source code and explanation:

Square Grid

Schwarz–Christoffel mapping from the square grid to the unit disk.

Source code and more explanation:

“Gaussian random embedding of multigraphs”, by Jason Cantarella, Tetsuo Deguchi, Clayton Shonkwiler, and Erica Uehara:

I’m very proud of this one: we give a principled mathematical model of topological polymers of arbitrary complexity which both recovers the standard physics model and is simple to both compute and prove theorems with.

Rotation Redux

A grid of circles, inverted in the unit circle, then mapped by the inverse Cayley transform.

Source code and more explanation:

Fourth Power

A square grid in the first quadrant under the map \(z \mapsto z^t\) as \(t\) ranges from \(1\) to \(4\), reflected across the line \(y=-x\) (plus some other stuff).

Source code and more explanation:


A Schwarz–Christoffel mapping of the upper half-plane to an equilateral triangle.

Explanation and source code:


Level sets of the real and imaginary parts of \(\frac{1}{\pi i} \log z\), mapped from the upper half plane to the unit disk by the Cayley transform.

Source code and more explanation:


\(z \mapsto \frac{4}{z}\) applied to an infinite stack of circles between the lines \(\{z:\operatorname{Re}(z)=1\}\) and \(\{z:\operatorname{Re}(z)=2\}\)

Source code and more explanation:

In which we show that the knots K13n592 and K15n41127 (pictured) both have stick number 10. These are the first non-torus knots with more than 9 crossings for which the exact stick number is known.


Made for the local Society for Industrial and Applied Mathematics (SIAM) chapter.

In which my student Tom Eddy and I give improved bounds on stick number of more than 40% of the knots up to 10 crossings for which it was not previously known.

Here's a 10-stick 10_16.


This is an animated version of a figure in my Bridges paper (, showing a geodesic in the Grassmannian \(G_2(\mathbb{C}^n)\) between a \(+3\)-framed regular 200-gon and a 0-framed regular 200-gon.

Source code and further explanation:

Off the End

Each row shows the stereographic projection of a rotating regular polygon to the line.

Source code and further explanation:


A pen plotter print showing orthogonal projections of the 13 Catalan solids.

Limited edition of 5.

Buy one here:

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