a webpage exploring various curiosities in knot theory; presented by Miao Lei, Kriteen Shrestha, Cindy Zhao, and mentor Marcos Reyes as part of UCSB DRP 2023
You may see knots when you're tying your shoe, or trying to untangle an old necklace. Loosely speaking, a knot is a connected and tangled loop of string with no thickness. Mathematically, a knot is a closed curve that is embedded in 3-D space.
The most basic knot is knot a knot at all (pun intended), but the circle, known as a loop, a round thing, or within knot theory, the unknot. One knot can be represented in different ways; those would be called different projections of a knot. Just as we do with numbers, we have prime and composite knots, which just mean that some knots cannot be reduced to any other knot, and some knots are composed of multiple knots.
these are pictures of the same knot.
The fundamental question we are tasked with answering is whether there exist distinct knots besides the unknot.
❓ Fundamental Question: Are all knots just different projections of the unknot?
The ongoing study of knot theory comes from finding different ways to differentiate the various knots. We use strategies called invariants to help us distinguish two knots. Like the name suggests, these strategies should not have a different outcome (remain “invariant”) for different projections of the same knot.
It is important to note that with invariants, rather than concluding that two knots are equivalent, we can only confirm whether two knots are not equivalent.
(the images below are produced from Mathematica’s Knot Data Set!)
Let’s look at how we can explore the fundamental way of transporting from one knot to another, using changes known as Reidemeister moves.
Reidemister moves are alterations that we can make to a knot to traverse from different projections of the same knot. In other words, a knot is equivalent, or ambient isotopic, if they are related through the following moves:
Therefore, if we can show a knot property holds after making any of the three Reidemister moves, we can conclude that the property is a knot invariant. That is, when applying an invariant on two knots, if the outcome of the two knots are the different, we can conclude they cannot be equivalent.
Note that if the outcomes are the same, we still cannot conclude that the knots are the same
For Reidemeister Move I below, the entire twist can be assigned a single color since it only includes two strands which means it cannot be colored by three distinct colors.
After Move I, we can see it can still keep the same color for the resulting strand.