The Universe Consists of Knots―A Key Idea on Geometry and Physics―

Abstract

Gauss studied knots and links, and introduced a word theory of plane curves. Tait gave a table of knots, and obtained a famous conjecture of knots. After Jones polynomial was appeared and this Tait conjecture was solved, we noticed an importance of studies of knot projections (plane curves). We introduce recent studies. From 1984, Jones polynomial, its generalization (Quantum invariants), and their categorifications have been hot topics in Geometry and Physics. From 2005, by Turaev’s nanoword theory, we can describe which information of a knot produces the Jones polynomial and the Khovanov homology. In this report, we introduce recent works of mathematicians and the author with respect to these topics.