Knotting and linking occurs in the tangling of ropes in physical space and the earliest human technologies utilized weaving and knotting in the making structures for containment, clothing and binding. Only since the time of Faraday, Maxwell and Kelvin, however, have knots begun to figure in physical and mathematical science. Faraday saw that electric and magnetic fields are literally linked with one another and further developments featured in the work of Gauss and Maxwell. From there Gauss then Riemann then Poincare developed the subject of topology and by 1900 it was possible to study knots using topology. A significant reconnection with physics occurred in the 1980’s work of Vaughan Jones connecting knot theory and statistical mechanics and with the work of Edward Witten connecting knot invariants with quantum field theory. Since that time there has been an extraordinary interplay of work relating knots and physics and natural science. Some of this work goes back to fundamental questions about knots as physical entities with length and thickness, elasticity and bendability. Some of it goes more deeply into the relationships with statistical mechanics, combinatorics, quantum field theory and new forays into algebraic topology undreamed of in the age of Faraday and Maxwell. Possible applications loom in quantum computing, molecular biology and elsewhere in pure mathematics and the sciences. This talk will be a pictorial introduction to these revolutions.
