August 5, 2024

Jessica A. Dennehy 

11th Grade

Williamsville East High School

After a decade passed, the vortex theory gained traction in the scientific community. Thomson gradually abandoned the vortex theory, focusing instead on other scientific accomplishments, including his international system of absolute temperature, named after his title, Lord Kelvin. Meanwhile, Tait was still heavily invested in these knots, convinced that the different varieties of knots could be tabulated into an equivalent of a table of elements. Although that was later proved false, his continued interest and research into knots ultimately established the mathematical field of knot theory. 

Although our knotted shoelaces and jewelry chains can be cumbersome, they don’t relate to knot theory. Knot theory applies exclusively to mathematical knots, which are categorized as simple closed curves within three-dimensional space. Therefore, what the general population would refer to as a ring is, in fact, a knot—it has a name: the unknot. An unknot, or trivial knot, is the simplest of all knots, as it can be drawn with no crossings. All mathematical knots are closed loops with no loose ends, but more complex knots have varying amounts of crossings. Hence, knot theory studies these closed curves and how they can be altered without cutting them. 

Knots are classified by properties that depend on the knot itself, regardless of its appearance at any given point in time. All knots can be repositioned to accurately display their specific properties, known as invariants. Some invariants include the minimal crossing number, which is the least number of crossings that can appear in any projection of the knot, and the unknotting number, which is the smallest number of “crossing changes” needed to turn the knot into an unknot. For reference, a trefoil knot, which has three points of crossover, has a minimal crossing number of three. In contrast, the unknotting number of a trefoil knot is one, as adjusting a crossing from under to over will morph the trefoil into an unknot. These “crossing changes” are formally known as Reidemeister moves, named after the German mathematician Kurt Reidemeister, who proved their use back in 1926. These movements—twisting/untwisting, crossing/uncrossing, and moving the physical strands—are essential in knot theory. 

Although knot theory developed as a purely mathematical discipline in its first century, major breakthroughs starting in the 1980s have linked it to a plethora of other subjects. In 1984, New Zealand mathematician Vaughan Jones introduced the concept of a new knot invariant, named the Jones polynomial. This advancement allowed Edward Witten, an American mathematical physicist, to discover a connection between knot theory and quantum field theory. Both were awarded Fields Medals in 1990 for their contributions to the field. A fellow Fields Medalist and American mathematician, William Thurston, made an impressive discovery linking knot theory with hyperbolic geometry, which has potential ramifications in cosmology. Further applications of knot theory have emerged in biology, chemistry, and mathematical physics. Although Tait’s research began with simple loops, the profound discoveries made through knot theory remind us that spinning in circles doesn’t mean progress isn’t being made.