History of knot theory

Knots have been used for basic purposes such as recording information, fastening and tying objects together, for thousands of years. The early, significant stimulus in knot theory would arrive later with Sir William Thomson (Lord Kelvin) and his theory of vortex atoms.

Different knots are better at different tasks, such as climbing or sailing. Knots were also regarded as having spiritual and religious symbolism in addition to their aesthetic qualities. The endless knot appears in Tibetan Buddhism, while the Borromean rings have made repeated appearances in different cultures, often symbolizing unity. The Celtic monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork.

Knots were studied from a mathematical viewpoint by Carl Friedrich Gauss, who in 1833 developed the Gauss linking integral for computing the linking number of two knots. His student Johann Benedict Listing, after whom Listing's knot is named, furthered their study.

In 1867 after observing Scottish physicist Peter Tait's experiments involving smoke rings, Thomson came to the idea that atoms were knots of swirling vortices in the æther. Chemical elements would thus correspond to knots and links. Tait's experiments were inspired by a paper of Helmholtz's on vortex-rings in incompressible fluids. Thomson and Tait believed that an understanding and classification of all possible knots would explain why atoms absorb and emit light at only the discrete wavelengths that they do. For example, Thomson thought that sodium could be the Hopf link due to its two lines of spectra.

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