March 1, 2023
In one of my research projects, my mentor, Dr. Carolyn Otto, and I are using different methods of combinatorics to construct 3-manifolds. At the moment we have begun focusing our study to lower dimensional objects, such as knots or graphs, and are planning to take these objects and embed them in an even higher dimension. Currently, we are examining knots and their different invariants. Knots have their own branch of mathematics, called knot theory, and it first gained exposure due to physicists believing it could have ties to the structures of atoms. Although these physicists weren't quite on the right track, knot theory proved to be a gold mine for mathematicians.
What is a knot?
Webster's dictionary describes a knot as "an interlacement of the parts of one or more flexible bodies forming a lump or knob". This description is not too far from what a mathematical knot is. The one key difference is that mathematicians work with knots formed on closed loops. So if you were to take a string and interlace the string to form a lump or knob and then glue the ends of the rope together, this would give us a mathematical knot.
We can think of mathematical knots as being physically knotted at all times. For example, when I tie my shoe I can always untie it, it is simply acting like a knot for some time. Now look at the picture above on the right, can you see any way to unknot it without damaging the string? Hopefully, the answer is no. The knot pictured above is called a trefoil knot and it happens to be the simplest knot that is not just a closed loop. A closed loop is called the unknot, pictured below, and although it has no tangles and does not act like a knot, it makes sense to include it in the set of knots. In a sense, the unknot is similar to the number zero.
Take a look at the two knots pictured below, one of them is the trefoil knot. Is the other a trefoil knot as well?
The left knot pictured is indeed the trefoil knot, but the right knot pictured is the unknot. Simply imagine lifting the large loop that is laying on top of the knot and untwisting it, then it becomes the unknot.
How do we work with knots?
The move that we used to untwist the knot above is one of three important moves in knot theory. These three moves are useful in the concept of knot equivalence. Knot theorists say that two knots are equivalent if one knot can be wiggled about smoothly in space without intersecting itself until it coincides exactly with the other knot. This type of wiggling can be classified with three moves, known as Reidemeister moves and typically called R1, R2, and R3. They are as follows:
R1: We can untwist or twist a loop.
R2: We can pass one string over another string to form two crossings or undo this same move.
R3: We can slide a piece of string over a crossing.
Using these moves we can tell if a knot is just another knot in disguise. Reidemeister moves help us determine knot invariants and help classify knot types. The classification of types of knots has the potential to help in our research as we embed different knots into higher dimensions.