On June 30, 2025, Mark Brittenham and Susan Hermiller uploaded a preprint to the arXiv called “Unknotting number is not additive under connected sum” (and an updated version on September 15, 2025). In it, they surprised the mathematical community by giving a counterexample to a long-standing conjecture in knot theory. The story was picked up…
Tag: knot theory
Drawing a Celtic Knot
I decided to put together some activities for a gathering of math and computer science majors—puzzles, math crafts, etc. One of them was to draw a Celtic knot. I decided to write up a handout with the instructions. I thought I’d post it here. Enjoy! Here’s a printable pdf of the handout and the diamond-shaped…
A game for budding knot theorists
Thanks to Sam Shah for introducing me to this fascinating online game: Entanglement. The rules are simple. You are given hexagonal tiles, one at a time, each adorned with six short segments of rope. Use them to construct the longest possible knot (measured in segments) before running into a wall. Entanglement is fun and addicting!…
Mathematical magic tricks for kids
My six-year-old son loves the website ActivityTV.com, especially their science, origami, cooking, and magic videos. I watched a few of the magic how-to videos with him and was pleasantly surprised to see that some of them had a distinctly mathematical feel to them. For example: Jumping rubber bands: topological properties of circles and linked circles…
Using wikis in mathematics classes
Wikipedia describes a wiki as a website that allows the easy creation and editing of any number of interlinked web pages… [Wikis] are often used to create collaborative websites, to power community websites, for personal note taking, in corporate intranets, and in knowledge management systems. I have used wikis in three of my classes: two…
Kindergarten mathematics
This is a call for help. My son’s kindergarten teacher has invited parents to come in and talk about their careers. I’d like to go in and talk about math. I’d like to have some interactive hands-on mathematics activities for the kids to do. I also want them to be activities outside the typical kindergarten…
The maypole braid group
Over the weekend I attended a May Day party thrown by one of my colleagues. During the party they had a traditional maypole dance. An example of a maypole dance is shown at left. A maypole is a tall pole with colorful ribbons attached to the top that are fanned out in a cone shape….
Bubbles with knotted boundaries
We can think of a mathematical knot as a knotted piece of string (or in our case, wire) with its free ends joined. Examples are shown below. There is a remarkable theorem that every knot can be realized as the boundary of a surface. Moreover, Herbert Seifert produced a very simple algorithm for constructing an orientable…
How to use KnotPlot
As I’ve mentioned before, I’m teaching a knot theory class this semester. I’ve been playing around with KnotPlot, a powerful piece of software for drawing and working with knots. I want my students use it, but it has a somewhat unintuitive interface. So I’m trying to write up a list of easy-to-use instructions for them. The…
Materials for a knot theory class
This is a call for help—or for suggestions, at least. I’m teaching a knot theory class next semester. I’m looking for good props to use in the class to make knots. I would like to be able to make knots such as the following (and have my students do so as well). I suppose the…
David Richeson: Division by Zero 