Knot a Problem — Investigation Ideas
Quick Celtic Knots
One of the points of this app was to be able to quickly draw Celtic Knot diagrams to make it easy to investigate their properties. While it is fun to draw your own, to study them you need a way to draw lots of them.
Here are some ideas of things to investigate. With all of them, start by looking for the pattern and then try to find an explanation.
- Start with a knot with no walls and investigate what happens when changing the size. What is the relationship between the size and the number of ribbons?
- Set the size so that there is one ribbon, turn the symmetry forcing off, and turn the orientations on. What happens when you add a wall that goes with the orientation? What if you add a wall in the opposite way?
- What happens to the number of ribbons when you add or take away a wall from a place where there are two ribbons?
- Can you make any predictions as to what happens to the number of ribbons when adjusting walls symmetrically?
- Can you make a pattern with three ribbons in which the first always crosses over the second, the second always crosses over the third, and the third always crosses over the first? What about with four ribbons?
There is a lot of Mathematics that can be drawn out of studying Celtic Knots. The area of Mathematics that they fit into is Knots and Links. The difference between a knot and a link is the number of ribbons: one ribbon is a knot, more than one is a link. Furthermore, Celtic knots and links are always alternating (can you explain why?).
- Calculate the writhe of a Celtic knot.
- Calculate the linking number of a Celtic knot.
- Which prime knots can you make?
Of course, one of the things about Celtic knots is how they look. The knots generated by this app have a reasonably fixed appearance, but it can be used as a starting point for designing more intrigate diagrams, either by using the TeX exporter or by drawing the knot again by hand. Below is a video of how to do this on squared paper.