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

TL;DR

This paper traces the historical development of knot theory from ancient times to modern categorification, introducing new homology theories based on Yang-Baxter operators and their potential for knot invariants.

Contribution

It introduces a novel homology framework for Yang-Baxter operators, extending Khovanov homology and connecting it with classical and modern knot invariants.

Findings

Homology of distributive structures generalizes Fox colorings.

Yang-Baxter homology offers new knot invariants.

Connections between homology theories and co-cycle invariants are supported.

Abstract

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