Laocoön’s sea serpent knot

In the middle of the Pacific ocean, an Ashley knot on the horizon. Maybe, Laocoön’s sea serpent on his way to Troy followed by two bundles of squids? Greeks love calamari!

Actually, this knot was developed by American knot expert Clifford Ashley in 1910. It was inspired by a stopper, a kind of a figure 8 knot, he saw on an oyster fishing boat. Add a sprinkle of Gauss code on the knot and that’s what it transforms into.

The Gauss code represents a knot with a sequence of integers. However, rather than every crossing being represented by two different numbers, crossings are labeled with only one number. When the crossing is an overcrossing, a positive number is listed. At an undercrossing, a negative number.

The ocean is an endless source of inspiration!

More on Ashley knot and the Knot-geometry project on Patreon



Unslicing Conway knot

This one in celebration of maths student Lisa Piccirillo  who recently unsliced the famous Conway knot that puzzled mathematicians for decades.

In knot theory, the Conway knot is a particular knot with 11 crossings. It is named after mathematician John Horton Conway

Genius mathematician Conway, who passed away recently, was also a great humanist. He would have loved to thank Lisa in person, I’m sure. 

#knot #knot geometry #Convay #Piccirillo #Kinoshita. More on the Conway – Kinoshita-Terasaka knot mutation @



Satellite knot

I have a great admiration for sailors. Day or night, just the ocean and the sky to look at. 

And then it happens!

Now, a satellite train keeps circling the planet in an endless loop. Is there any link between a satellite knot and a Starlink satellite train? None that we know of, but it’s a great segue for this week’s illustration. The combination of the two creates an intriguing design, mysterious, fearsome, and ethereal at the same time. 

Are we chain linking the planet?  Adrian Jimenez Pascual presented a new family of knots called lassos to construct families of satellite knots. Could they rope the pesky little balls too?


  • Center image: A train of Starlink satellites passing over Pukehina, New Zealand. Image courtesy Astrofarmer.



Braid, symmetry, and quantum physics

A bead curtain? At first, nothing exceptional.

Actually it is a 21 string braid made of 1033 beads. Sailing this week in the middle of the Pacific, with 90% humidity , vapor lines must be raising like hazy braided curtains, a good segue for the Knot-Geometry project!

In mathematics, a braid is an intertwining of some number of strings attached to top and bottom “bars” such that each string never “turns back up.” This arrangement belongs to the Artin braid group and display a symmetry that can expand to infinity.

A little like the background, a simple view of the ocean, the sky and some clouds. I converted a generic view of the ocean into a more abstract figure. The original is included in my Patreon portfolio, wind map #18.

More on Patreon and the knot geometry project @


Of spirals and cyclones

Cyclones. They can be messy. They are characterized by spiral patterns.

Spirals, going back to the beginning of time,  can be found in every culture, worldwide. Was the symbol created out of fear? Out of respect? We’ll never know. 

However, the mathematical spiral knot,  the trefoil knot – cousin of the Triskelion, soars like an elegant signature, or maybe the ghost of a weightless dancer twirling and turning on itself.

Simple, minimal and graceful, the spiral knot can also signal chaos and destruction in the order of things. However, unlike many knots, the trefoil knot get all unknotted in the 4th Dimension. Just like cyclones – they appear, grow, create havoc and die out.

An intriguing connection between spirals, spiral knots, cyclones and cultural symbols.


The knot puzzle

Leaving the Galapagos behind and moving westward, 5,000 (nautical) knots to the next above-water land. What better time to start that leg of the Knot-geometry journey than playing a game called knot puzzle created a few years ago by Akio Karachi & Kendo Kishimoto, two mathematicians from Osaka U.

The puzzle ask me to identify each region and line up the crossing sections in a patch of unified bright color disks.

The knot puzzle I selected is closely related to the 6.2 crossing knot one of 3 prime knots with six crossings. It is alternating, chiral, and invertible.

In the third dimension, this knot is associated with the Borromean rings. Composed of three interconnected rigs, it creates a perplexing figure as no two of the rings are linked with each other, yet all three are linked together.

The open ended version of the 6.2 knot, the stevedore, or docker knot can be found on many shipping docks around the world.

Culturally, it figures in Kolam, an Asian geometrical line drawing tradition drawing rice powder pattern on the floor of the house to invites birds and other small creatures to eat it, thus welcoming other beings into one’s home and everyday life: a daily tribute to harmonious co-existence.

From the House of Borromeo to Indian floors and shipping docs,  this knot demonstrates quite unusual but very versatile qualities.


The turtle knot

Week 15 of the knot geometry project. An 8-15 knot to celebrate the 15 subspecies of the Galapagos giant turtle according to Academy herpetologist John Van Denburgh.

Standing on a 8.15 braid-shaped iron work balcony this invertible knot rests on what is an actual giant turtle shell. 900 lb or more, the Galápagos giant tortoises used to populate the entire planet but today can only be found in two small islands, the Galapagos being one of them.

More on the Knot-geometry journey, wind maps and mathematical knots on Patreon