Mathematical knots
Mathematical knots are a fascinating area of study that blends concepts from topology, physics, and biochemistry. In topology, a knot is defined as a loop in three-dimensional space that is tangled, with the simplest example being the unknot, which resembles an untangled rubber band. The field of knot theory seeks to understand the properties and classifications of these knots, particularly through a series of manipulative operations known as Reidemeister moves. Mathematicians utilize various invariants, such as the Alexander polynomial and the Jones polynomial, to differentiate between knots, although some knots can share the same invariant values.
In addition to their theoretical significance, mathematical knots have practical applications in physics, particularly in topological quantum field theory, where they represent interactions between particles. Furthermore, knot theory plays a crucial role in biochemistry, especially in understanding the structure of DNA. The knotted formations of DNA can impede replication, and specific enzymes, called topoisomerases, are essential for managing these knots during the replication process. Overall, the study of mathematical knots bridges abstract mathematical concepts with real-world applications, making it a rich and dynamic area of research.
Mathematical knots
SUMMARY: Mathematical knots are useful in physics and biochemistry.
Since ancient times, knots have been used in sailing, building, textiles (“knit” comes from “knot”), climbing, and in recreation, as well as serving as symbols for spiritual or religious concepts like eternity or wisdom. Topology generalizes the idea of a knot to an embedded circle in 3-dimensional Euclidean space. In knot theory, a knot is a tangled-up loop, like a piece of string with the ends fused together. The simplest is the unknot, simply an untangled loop like a rubber band. Two knots are the same if one can be manipulated (transformed) into the other without breaking the loop or passing the string through itself. In 1926, Kurt Reidemeister demonstrated that all such transformations were made up of a sequence of just three basic moves called Reidemeister moves. Deciding whether two knots are the same via a sequence of such moves is a member of a host of problems involving changing one object into another without breaking or tearing, which have long stumped topologists. Topologists find it difficult to assure themselves that failing to transform one knot into another truly reflects impossibility, or rather just their own failure. In modern times, mathematical knots are useful in physics and biochemistry.
Invariants and Links
To wrestle with this problem, topologists have created an assortment of invariants, mathematical entities that can be unambiguously computed for each knot. If a particular invariant has different values on two knots, then those knots are different. Unfortunately, different knots can have the same invariants. In 1928, James Waddell Alexander II created a method for associating a polynomial to a knot, now called its Alexander polynomial. In 1983, Vaughan Jones, studying a simplified model of phase transitions, such as freezing, discovered a second invariant, the “Jones polynomial.” Another mathematician, Edward Witten, soon noticed that the same polynomial could be computed from an invariant on particular three-dimensional spheres, providing insight into another difficult classification problem. Witten and Jones shared part of the Field’s Medal in 1990 for these discoveries. Victor Vassiliev has since created a host of new invariants. The Vassiliev invariants are infinite in number, and it is conjectured that any two different knots will differ in at least one such invariant.
Not all invariants are polynomials. Henri Poincaré created a topological invariant called the “fundamental group.” Applied to knots, it is called the “knot group” and is actually computed on the complement of the knot, that is, the abstract concept of all space with the knot removed. Poincaré’s invariant was the seed of an area that grew into a central focus of twentieth-century mathematics called “homological algebra.”
Knots, and their close cousins, links, have proven useful in a branch of physics called “topological quantum field theory.” For this application, physicists use particular guidelines to trace knots in two dimensions. The knot diagrams then portray scenarios in which particles are created, interact, and are finally annihilated. By appropriately labeling pieces of knots, mathematicians can realize the Jones and other invariants via important modern mathematical constructs, including the Yang–Baxter equations and quantum groups. Mikhail Khovanov, a Russian-American professor of mathematics at Columbia University in New York, has created a new type of invariant on links known as the Khovanov homology, which is a categorification of the Jones polynomial. This topic remains at the forefront of contemporary mathematics.
Applications in Biochemistry
The application of knot theory to DNA molecules has helped to elucidate their biochemistry. The DNA molecule of a bacterium closes into a circle, which bends and twists itself into a knot. This knotted structure can block DNA replication. Using electron microscopy or gel electrophoresis, the biologist can determine an individual molecule’s crossing and unknotting numbers, two numbers that classify knots. Enzymes called “topoisomerases” release the knots as a preliminary step to DNA replication. By carefully examining the knots that arise, molecular biologists have determined that there are two different topoisomerase molecules. Topoisomerase I releases the knot by cutting both strands of the molecule, and Topoisomerase II nicks just one strand and twists the cut strand around the other.
Bibliography
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