A Comprehensive History of Knot Theory: From Lord Kelvin to Witten's Breakthrough
Knot theory, a branch of mathematics that studies the properties of knots and their embeddings in space, has a rich and fascinating history. Its origins can be traced back to the mid-19th century, when Scottish physicist and Nobel laureate Lord Kelvin proposed the idea that atoms were knotted vortex tubes of ether. This groundbreaking concept, while ultimately incorrect, marked the beginning of a significant area of mathematical inquiry that continues to this day.
Early Beginnings and the Introduction of Knot Theory
The story of knot theory begins in 1867 when Sir William Thomson (known as Lord Kelvin at the time) proposed a theory that would rewrite our understanding of the fundamental structure of matter. Thomson suggested that atoms were composed of knotted vortex rings of a hypothetical energy field, ether. This notion of atoms as knotted structures was a bold and imaginative hypothesis in the field of physics, and it sparked a wave of interest in the mathematical properties of knots and their potential applications.
P. G. Tait and the Early Classification of Knots
Peter Guthrie Tait, who was Kelvin's collaborator, became the first to delve into the mathematical classification of knots. His work laid the foundation for the study of knot theory, and his discoveries, known as the Tait conjectures, are still important references in the field. Tait strived to provide a rigorous mathematical framework for understanding the structure of knots, a task that was crucial for the development of knot theory as a distinct mathematical discipline. Despite the inaccuracies of Kelvin's atomic model, Tait's work has had a lasting impact on the field.
Progress and the Discovery of Link Invariants
The field of knot theory took a significant turn in 1928 with the discovery of the first link invariant polynomial: the Alexander polynomial. This discovery by James Waddell Alexander II was a crucial step in the development of topological invariants that could be used to classify knots. However, while the Alexander polynomial provided a useful tool, it was not powerful enough to prove all of the Tait conjectures, as these conjectures required more sophisticated invariants to be fully addressed.
The Emergence of the Jones Polynomial
A major breakthrough in knot theory occurred in the 1980s with the discovery of the Jones polynomial. In 1984, Vaughan Jones, a mathematician, introduced the Jones polynomial, which became a cornerstone of modern knot theory. Unlike the Alexander polynomial, the Jones polynomial was more versatile and powerful, making it possible to prove several of the Tait conjectures. This development marked a significant turning point in the study of knots and their applications.
The Reintroduction of Knot Theory to Physics: Witten's Contribution
After a period where knot theory was largely ignored, it experienced a resurgence in the late 1980s with the work of Edward Witten, a mathematical physicist. Witten's contributions not only led to a deeper understanding of knot theory but also opened up new avenues in theoretical physics. Building on Michael Atiyah's proposal, Witten showed that the expectation values of the Wilson loops in Chern-Simons theory could be described by the Jones polynomial or related invariants. This discovery had profound implications for gauge theories and conformal field theories, and it demonstrated the profound connection between knot theory and modern physics.
Conclusion
In conclusion, the history of knot theory is a testament to the enduring power of mathematical creativity and its potential to impact various fields, from classical physics to modern theoretical physics. The journey from Lord Kelvin's atomic theory to Witten's groundbreaking work showcases the evolution of knot theory from a novel mathematical concept to a powerful tool in understanding the universe. As our understanding of knots continues to grow, so too does the potential for new discoveries and applications in science and mathematics.