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This paper is an extended account of my "Introductory Plenary talk at Knotsin Hellas 2016" conference We start from the short introduction to Knot Theoryfrom the historical perspective, starting from Heraclas text (the first centuryAD), mentioning R.Llull (1232-1315), A.Kircher (1602-1680), Leibniz idea ofGeometria Situs (1679), and J.B.Listing (student of Gauss) work of 1847. Wespend some space on Ralph H. Fox (1913-1973) elementary introduction to diagramcolorings (1956). In the second section we describe how Fox work wasgeneralized to distributive colorings (racks and quandles) and eventually inthe work of Jones and Turaev to link invariants via Yang-Baxter operators, herethe importance of statistical mechanics to topology will be mentioned. Finallywe describe recent developments which started with Mikhail Khovanov work oncategorification of the Jones polynomial. By analogy to Khovanov homology webuild homology of distributive structures (including homology of Fox colorings)and generalize it to homology of Yang-Baxter operators. We speculate, withsupporting evidence, on co-cycle invariants of knots coming from Yang-Baxterhomology. Here the work of Fenn-Rourke-Sanderson (geometric realization ofpre-cubic sets of link diagrams) and Carter-Kamada-Saito (co-cycle invariantsof links) will be discussed and expanded.  Dedicated to Lou Kauffman for his 70th birthday.

Knot Theory: from Fox 3-colorings of links to Yang-Baxter homology and Khovanov homology