Knot theory

Lore & Background

The first systematic mathematical study of knots is generally attributed to Carl Friedrich Gauss and later Peter Guthrie Tait. In the 19th century, Gauss defined the linking integral, and Lord Kelvin's theory that atoms were knots in the aether led Tait to create the first knot tables for complete classification. Early 20th-century topologists such as Max Dehn and J. W. Alexander studied knots using knot groups and invariants like the Alexander polynomial.

Reader's Guide

Knot theory has evolved from a 19th-century attempt to classify knots into a rich mathematical discipline with deep connections to geometry, physics, and biology. In the late 1970s, William Thurston introduced hyperbolic geometry, showing many knots are hyperbolic and enabling new invariants. In recent decades, knot theory has been applied to study chirality in molecules, the action of topoisomerase on DNA, and the construction of quantum computers through topological quantum computation. More than six billion knots and links have been tabulated since the 19th century. The field continues to address fundamental problems, such as determining when two descriptions represent the same knot—a problem with a known algorithmic solution of unknown complexity.

Did You Know?

• A mathematical knot differs from everyday knots in that its ends are joined so it cannot be undone; the simplest knot is a ring called the unknot.
• Over 6 million prime knots with up to 20 crossings have been tabulated since the beginnings of knot theory in the 19th century.
• Knot theory can be used to determine if a molecule is chiral (has a 'handedness') or not.