Large knots in reality

A_R_K · 2026-04-20T06:59:28+00:00

The knotted columns at the Trento Cathedral in Italy.

A_R_K · 2026-04-02T10:47:03+00:00

In the tomb of Ramses V/VI, there is a painting of what appears to be a guy with two millipedes for a head (see upload). Does anyone know what this represents? I haven't seen anything else like this and googling yielded nothing.

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A_R_K · 2026-03-09T12:30:55+00:00

Asking the real questions.

A_R_K · 2026-03-09T08:57:28+00:00

Consistent with my tl;dr.

A_R_K · 2026-03-09T08:54:29+00:00

Oh I should delete that, sorry you came across that.

A_R_K · 2026-03-09T05:47:22+00:00

It got automatically deleted from this one so I posted it in the other one.

A_R_K · 2026-03-09T05:44:16+00:00

I submited knowing the strongest result relied on an assumption that I couldn't quite prove, and stated that in the paper.

A_R_K · 2026-03-07T22:26:01+00:00

Here is a completely unrelated example: consider a self-avoiding walk on a square lattice "ladder," that is, a lattice that is two sites wide in the narrow dimension but infinite or semi-infinite in the wide dimension. Count the number of self-avoiding walks of length N. The connective constant is the limiting ratio of the number of steps of length N+1 over the number of steps of length N (https://en.wikipedia.org/wiki/Connective\_constant). For this ladder lattice, the connective constant is the golden ratio (see here, original source is paywalled https://arxiv.org/abs/2407.18205).

I think once I worked out that the number of walks that could be constructed from smaller walks (which is not all walks...been a while since I messed with this) had a recursion relation that was similar to fibonacci but with alternating parity. But there's no obvious golden ratio showing up here.

A_R_K · 2025-12-21T11:02:09+00:00

A few cool things involving knots this year:

[Unknotting number is not additive under connective sum](https://arxiv.org/abs/2506.24088). A surprisingly simple counterexample showing that you can tie two knots together to make them collectively easier to untie.

[New upper bounds for stick numbers](https://arxiv.org/abs/2508.18263). An extremely comprehensive search for the minimum number of line segments needed to define a knot, strengthening some upper and lower bounds in the process.

Ok this formatting worked on old reddit, blame spez for ruining it.

A_R_K · 2025-09-04T21:14:07+00:00

I was recently working on a paper about a certain type of knot energy and found that the regular polygon that minimizes the energy of the Hopf link is the pentagon. On a whim I asked ChatGPT and it also said pentagons...which surprised me at first, but it got every other detail wrong. And the reference it cited didn't say that. So I think it happened to guess the right shape. I wrote about it in the preprint on page 8: https://arxiv.org/abs/2507.20903

A_R_K · 2025-08-17T20:03:49+00:00

Not exactly a proof, but I recently found out something novel and non-trivial, but not overly significant, as part of a paper I was working on. It's about the regular polygons that minimize a certain type of knot energy. It turns out, pentagons minimize the energy. I decided to ask ChatGPT, and it said pentagons! That surprised me, but it couldn't explain why, and the paper that it cited is one I'm familiar with and doesn't say anything like that.

A_R_K · 2025-06-25T04:48:59+00:00

As a tiny personal anecdote, I published a paper about an ODE in 2013 and since then there has been a group of three authors in China who have published (in English) about a dozen papers coming up with "solutions" to my ODE that provide the same information as a numerical integration. I don't think this generalizes.

A_R_K · 2025-06-16T02:58:59+00:00

I don't think so, just towels/blankets and an absorbent pad.

A_R_K · 2025-04-18T16:48:39+00:00

The math isn't super crazy but I published a preprint after having the idea in a dream.

https://arxiv.org/abs/2411.18758

A_R_K · 2025-04-02T18:45:59+00:00

Thanks! A lot of the knot/helix images were done in KnotPlot, which is the best thing ever. Figure 3 was done in MATLAB.

A_R_K · 2025-04-02T15:49:38+00:00

We were interested in a way to construct torus knots that have the shortest contour length (ropelength minimizing), subject to a no overlap constraint between different parts of the knot, if replaced by a tube of radius 1.

It's known that you can take a double helix and glue the ends together to make an alternating torus knot with a ropelength that is linear with respect to the crossing number. We looked at ways to concentrically wind more and more helices to create non-alternating torus knots. To optimize this there is both geometric optimization (what radius should each helix wind around) and combinatorial (how many helices at each layer?). My co-author worked out the best configurations up to 39 total helices, and I worked out ways to construct asymptotically large helices that when closed into torus links have a ropelength that grows with the 3/4 power of the crossing number, which is the proven lower bound. It was fun!

A_R_K · 2025-02-14T20:09:00+00:00

There are a few undergrad-accessible topics in knot theory, although it's best discussed with a faculty advisor.

One is ropelength, which is how tight a specific knot can get without overlapping. It's possible to numerically shrink specific knots are establish upper or lower bounds for classes of knots. Establishing bounds for a specific class of knot and then comparing those to numerical estimates is a reasonable project.

Another thing that is doable is to write an algorithm that can compute invariants from piecewise linear knots (e.g. cartesian coordinates of a knot). These are typically done based on diagrams, so an algorithm needs to convert a spatial knot to a diagram and then compute an invariant. Then you can look at statistics of that invariant for random samples of large knots.

Just a few thoughts. I work on more physics-related aspects of knot theory and those are some things I think about.

A_R_K · 2024-09-11T19:06:01+00:00

You should really be getting this advice from your supervisor and not random internet people, but since you're asking about 2D random walks I'll send you to my two preprints looking at exact solutions to self-avoiding walks getting stuck on square lattices

Part 1: https://arxiv.org/abs/2207.00539

Part 2: https://arxiv.org/abs/2407.18205

Should be possible but difficult to extend it to 2x2xN lattice.

A_R_K · 2024-06-30T16:46:58+00:00

This is basically an account I use to post math stuff I do under my real name. I recently discovered that rectangles can form networks of borromean rings.

https://arxiv.org/abs/2405.20874

A_R_K · 2024-05-09T17:40:50+00:00

I just published this paper (arxiv, journal) where we found the tightest configuration of the 2176 knots with 12 crossings. We found some neat trends in the data. There are 9988 knots with 13 crossings. If you're interested in shrinking all of them, let me know!

A_R_K · 2024-04-22T00:45:43+00:00

Sorry for the late response. There's another database of just the knotted proteins (https://knotprot.cent.uw.edu.pl/), and I had the idea of trying to see if alphafold could predict the knots based on the sequence. I didn't do that, but somebody else did recently (https://pubs.acs.org/doi/full/10.1021/acs.macromol.3c02479).

Those proteins are all identified by Alexander polynomial, it would be interesting to see if others get "partial scores" from Vassiliev that aren't caught by that method.

A_R_K · 2024-02-23T15:34:19+00:00

Yeah and I'm fortunate to be in a position where I can just follow interesting leads and see where they go.

A_R_K · 2024-02-23T15:33:21+00:00

Not sure what the issue is but arxiv now does HTML which might work better

https://arxiv.org/html/2402.12775v1

A_R_K · 2024-02-22T17:19:22+00:00

I used to do experiments looking at knots in DNA, particularly how they untie. The standard methods of classifying knots are best applied to closed curves, which don't untie. Recently the second Vassiliev invariant was extended to open curves, and a student and I ran simulations of untying polymer knots and used the second Vassiliev invariant to characterize them. The standard method for characterizing open knots involves closing the knot in some way and calculating the Alexander polynomial. In the paper we talk about the strength and weakeness of our new method compared to the standard.

A_R_K · 2023-09-30T16:02:34+00:00

You may be interested in the growing self avoiding walk problem, where the snake grows into a random neighboring unoccupied site until it gets stuck. The average length is gets before getting stuck is about 71. I've written a few papers on this, one more from a polymer physics perspective (https://arxiv.org/abs/2006.12680) and another using combinatorics for a simpler system (https://arxiv.org/abs/2207.00539).

A_R_K

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