Knot Theory Puzzles
At the Joint Math Meetings, some folks from the VCU Breakthroughs team presented some activities on knots using a silicone necklace manipulative. Today we'll look at a set of puzzles inspired by that.
Setup
For this activity I'll be using a manipulative that I got at Michaels craft store: Creatology silicone necklaces and bracelets.
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A pack contains \(6\) larger necklaces and \(6\) smaller bracelets for \(\$4\). The necklaces are easier to work with, so that's what I'll use. Some other options:
Silkies necklaces (6 for $6)
Waxed necklace cords (200 for $18)
The gist of this activity is identifying when one knot is the same as another. While this is a deep and central question in knot theory, students will interact with it experimentally and physically. Colloquially, to a mathematician, a knot is an arrangement of a string with both ends fused together. Two knots are equivalent if one can be turned into the other without needing to cut and re-fuse it.
Because a premade knot will twist around on its own while bouncing in a bin or pocket, the "resetting" mechanism for this activity is making a fresh knot by looking at a diagram. This probably sets the floor somewhere around mid-late elementary school just for patience and fine motor skills, but the concept is not particularly challenging.
Challenges
Below are three knots. Make each of them using your necklace, paying careful attention to the crossings. See if you can turn one into another without unclasping your necklace. After experimenting, do you believe that these three knots are different?
You have actually seen the six knots below before -- they are just the three knots above in disguise! Try to match each of the six knots below with its equivalent knot above. You might be able to tell just by looking, but you'll probably want to make the knots below with your necklace and try to twist them into the knots above.
While it might appear that there are five different knots below, there are actually only two! See if you can sort the knots below into groups of equivalent knots.
Try to organize the nine knots below into groups of equivalent knots. How many groups did you need?
The crossing number of a knot is the minimum number of times it can cross over itself when you lay it flat on the table. Try to make a knot with crossing number \(7\).
Can you find other knots with crossing number \(7\) not equivalent to the one you've found? How many others can you find?
Notes
The crossing numbers of the first three knots are \(0\), \(3\), and \(4\). These knots are called the unknot, trefoil, and figure-8 knot, respectively. After wrestling with them for a bit, students will likely at some point have them in a position to exhibit that number of crossings, which provides a nice distinguishing visual clue.
Some knots are chiral in the sense that they are not equivalent to their mirror images. For example, the trefoil is chiral while the figure-8 is not. I've tried to be careful to only include one member of each chiral pair whenever this occurs.
Two knots can be combined into a single knot by cutting them open and chaining them in a cycle. This new knot is the knot sum of the two other knots. For example, two trefoils of the same chirality give a granny knot while two trefoils of opposite chiralities give a square knot. Knots that cannot be decomposed into a knot sum of two other knots (ignoring the unknot) in this way are prime.
In the \(5\)-knot challenge, there are two copies of one knot and three of another. These are the only two knots with crossing number \(5\), not counting mirrors. In the \(9\)-knot challenge, there are three diagrams each of three distinct knots. These are the only three prime knots with crossing number \(6\), not counting mirrors.
Here are all prime knots with crossing number \(\leq 7\):
If students find a knot with crossing number \(7\), it must be one of the seven prime knots with crossing number \(7\), a mirror of one of those, or a composition of a trefoil or its mirror and figure-8.
Extra Challenges
Students might like to make a random knot and then try to classify it. Chances are it will be one of the knots in the table (or a mirror) or a knot sum of knots in the table (or mirrors).
You might have noticed there are no knots with crossing numbers \(1\) or \(2\) in the table above. Arguing why there can't be is an interesting exercise.
Our manipulative makes exploring knot sums straightforward. For example, here is a square knot:
Students can use this manipulative to explore the key idea behind the proof that knot sums are commutative. For example, if you start with yellow trefoil, red figure-8, blue trefoil, green figure-8, a nice challenge is to turn this into yellow trefoil, red trefoil, blue figure-8, green figure-8.
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