Thursday, September 27, 2018

some knots and not knots

Here are some knot patterns made using the online tool mentioned a few posts back. For each knot, the 'grid pattern' used to create the knot in the editor is also shown. First up, the simple unknot.

not a knot

Solomon's Knot
Next to this, we have a nice motif that is not a knot at all: the Solomon Knot is the name given to this motif of interlocked chains.

solomon unknot

BTW, you can make pattern that looks like  a solomon knot using truchet tiles:

truchet solomon unknot

Trefoil Knot
The trefoil is the simplest actual knot we can draw, but there are several ways to draw visually different but essentially equivalent trefoils, the first is considered the 'foundational' celtic knot:

trefoil 1

Here's another rendition:

trefoil 2

And one that is less recognisable:

trefoil 3

The Josephine Knot
Another non-knot (but very decorative), the Josephine consists of two interlaced links, and seems to be a favourite among crafters (who use it in belts, bracelets and macrame):

josephine link

Figure Eight Knot
The Trefoil is the only mathematically distinct prime knot with 3 crossings in a minimal planar diagram, and the figure 8 is the only one with 4. This version illustrates its name nicely:

figure eight 1

And here it is, a little lopsided:

figure eight 2

The Three-Twist
One of only two prime knots with five crossings, the three-twist (or 5_2) looks a lot like the figure 8. You can see how removing one of the lines in the grid pattern of the figure 8 allows for the additional twist that forms this knot.

the three-twist

The eight-eighteen
I don't know of a common name for the 8_18, but it's a nice looking knot, so it should have one. Here are two presentations of it:

eight-eighteen 1

eight-eighteen 2

Experiment with these and others here.