knots

History

Knot theory emerged in the 19th century when physicist Lord Kelvin hypothesized that atoms are vortex knots in the ether. Mathematician Peter Guthrie Tait began compiling tables of knots. Later, Henri Poincaré gave a rigorous topological definition on which modern science is built.

How it works

To distinguish one knot from another, invariants are devised — characteristics that do not change under bending. For example, the minimum number of crossings: the trefoil has three, the figure-eight has four. An invariant is like a unique passport number of the knot.

History

Knot theory emerged in the 19th century when physicist Lord Kelvin hypothesized that atoms are vortex knots in the ether. Mathematician Peter Guthrie Tait began compiling tables of knots. Later, Henri Poincaré gave a rigorous topological definition on which modern science is built.

How it works

To distinguish one knot from another, invariants are devised — characteristics that do not change under bending. For example, the minimum number of crossings: the trefoil has three, the figure-eight has four. An invariant is like a unique passport number of the knot.

History

The birth of knot theory is associated with the hypothesis of William Thomson (Lord Kelvin) in 1867, who proposed that chemical elements are knots in the ether. Peter Guthrie Tait began compiling tables of knots, classifying all knots up to 7 crossings. With the discovery of the electron and the abandonment of the ether, knot theory receded into the background, but at the beginning of the 20th century, through the efforts of Henri Poincaré and other topologists, it received a rigorous mathematical foundation. In the second half of the 20th century, a renaissance occurred thanks to the discovery of the Jones polynomial (Vaughan Jones, 1984) and connections with quantum physics, the works of Edward Witten and others.

How it works

To classify knots, invariants are used — quantities that do not change under isotopies. The simplest is the number of crossings in a knot diagram. More complex ones include the Alexander polynomial, the Jones polynomial (related to braid group representations and quantum invariants), Vassiliev invariants. The Jones polynomial is computed via skein relations: it is recursively determined through the relationship of knot diagrams that differ only in a small neighborhood of a crossing. In physics, topological knots arise as stable solutions in field theories; for example, in Maxwell's equations there exist electromagnetic knots whose stability is guaranteed by the Hopf topological invariant. World lines of particles in (2+1)-dimensional systems can form knots, leading to fractional statistics.

Open problems & fun facts

Complete classification of knots: there is no complete set of invariants that distinguishes any two non-equivalent knots.; Connection of quantum invariants with three-dimensional quantum gravity.; Efficient computation of invariants for complex knots, including cosmic and chromodynamic strings.

💡 The Jones polynomial allows distinguishing left-handed and right-handed trefoils, which is impossible with the Alexander polynomial.