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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:

School’s Out

100 random points in the first orthant on the unit sphere, undergoing two simultaneous random rotations and being stereographically projected to the plane.

Source code:

@christianp Not sure funding bodies would go for it, but an interesting hybrid would be for people to write, say, $500 or $1000 into their grants to go to arXiv.

@jsiehler Haversine is actually surprisingly useful if you want to smoothly interpolate between \(f(0)=0, f'(0)=0\) and \(f(a)=1,f'(a)=0\).

I also sometimes use the polynomial smoothstep/smootherstep/smootheststep functions:

Symmetric Minimality

A symmetric minimal lattice trefoil knot, inspired by a conversation here with @11011110, who asked about minimal lattice trefoils with more symmetry and ended up finding the coordinates of this one.

Source code:


Start with the vertices of the regular icosahedron on the unit sphere. Make circles on the sphere consisting of those points whose spherical distance to one of the vertices is no bigger than \(\frac{\operatorname{arcsec} \sqrt{5}}{3}\). Stereographically project to the plane and this is the result.

Buy it here:

@11011110 Yeah, that does seem like the right question, doesn’t it?


The Riemann mapping theorem guarantees a conformal map from the unit disk to any simply-connected planar domain; this shows an example of such a mapping to the equilateral triangle.

Buy it:

@11011110 Nice! That totally works! It’s in Diao’s class (c), but surprisingly it’s not the one he shows. It also looks to me like Scharein et al.’s 2–17 (, but they don’t remark on it either.

I wonder whether one should expect highly symmetric minimal lattice knots in general; my guess is probably not, but all the evidence seems to suggest that there are lots of minimal examples for each knot type.

@11011110 It’s entirely likely. I just took the one that’s built into KnotPlot, but there are 3496 minimal lattice trefoils (, so I wouldn’t be at all surprised if one is substantially more symmetric. Let me know if you figure it out!

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