The Mathematics of Knot Theory
When we think of knots, we normally visualize the ones we interact with daily, like when tying shoelaces or fastening ties. However, these knots differ from what is studied in mathematics, which are defined as closed loops that can be embedded in a three-dimensional space. This means that a mathematical knot cannot be untied or cut without its identity changing. Knots are so important that there exists an entire branch of mathematics dedicated to them, called knot theory. This field delves into the classifications and generalizations of these knots, as well as their various properties and applications. Knots appear in nature in a myriad of forms, such as the shape of a DNA molecule, the folding of proteins in our bodies, or the structure of a magnetic field. Therefore, by increasing our familiarity with knots, we can gain an insight into many of the natural phenomena that occur around us.
One of the most fundamental challenges of knot theory is the differentiation between two different knots; that is, whether they can be transformed into each other via a continuous deformation without breaking or crossing the loop. To accomplish this, knots are mapped onto diagrams that clearly identify their crossings which identify which strands go over and under one another at a point. This information must be relayed via a series of commonly used notations. For instance, some of the most commonly used notations include the use of a small gap between the lower strand and upper strands at a crossing point to indicate that it is below it. Another example is indicating whether a crossing is negative or positive by drawing a small circle with a plus or minus sign, depending on the orientation of the strands.
Two diagrams depict the same knot if one can be obtained from the other via a sequence of Reidemeister moves, which are changes that do not affect the type of knot. These moves are categorised into three types which are: twisting or untwisting a single crossing; sliding one strand over or under another; and moving a strand across a crossing.
There exist some knots, however, that are more complex than and thus cannot be made simpler by these Reidemeister moves. These knots are referred to as prime knots, and they are the building blocks of all knots. Any composite knot can be broken down into a product of prime knots via the connect sum operation, which is the act of cutting a small arc from each knot and joining them together.
Another challenge in knot theory is measuring the complexity or simplicity of a knot using several numerical invariants, which include the crossing number, bridge number, unknotting number, etc. These invariants are integers that remain the same no matter which Reidemeister moves are applied to a knot, and can therefore be used in the comparison or classification of knots. For instance, the crossing number is the smallest number of crossings in any diagram of a knot. The bridge number is the minimal number of bridges required in all possible bridge representations of a knot. The unknotting number of a knot is the least number of crossings that need to be moved from over to under and vice versa to obtain the unknot, a single closed loop. Specifically, the bridge representation of a knot is a diagram of that knot in which it lies mostly flat, except for certain sections that are lifted, or “bridged”, over the rest. Thus, the bridge number is the smallest number of these lifted sections that are needed to accurately represent the knot.
Other than the numerical invariants, there are also algebraic invariants of knots, which include the fundamental group, the Alexander polynomial, and the Jones polynomial. These invariants are more complex and are especially useful since they can differentiate between classes of knots that share the same numerical invariants. For example, the fundamental group of a specific knot is a group that describes how loops around the knot can be combined or inverted. The Alexander and Jones polynomials are polynomials that can encode some information about the crossings and orientations of a knot diagram. These polynomials have several remarkable characteristics and uses, such as having the ability to study the connections between knot theory and other fields like statistical mechanics and quantum computation.
Knot theory is not a field that is advanced for its own sake; it also enriches other areas of mathematics. For example, knots can be generalised to higher dimensions or other spaces, such as links, braids, tangles, etc. They may also relate to some other mathematical objects and structures, like manifolds, surfaces, groups, matrices, and more. These generalisations may lead to new results or insights in mathematics.
Moreover, knot theory also has applications outside of mathematics, especially in science and engineering. It may help us understand how enzymes can disentangle strands of linked DNA, how molecules with new and unique properties may be created, or how the interplay forces within the Sun’s corona have a knotted structure. Knot theory can also inspire new designs for neckties, shoelaces, logos, and art.
Overall, knot theory is such a crucial yet underappreciated field of mathematics that influences some simple yet important parts of daily life.