Quandles and Their Colorings of Knot Diagrams

Racks and quandles are fundamental algebraic structures. Quandles have been rediscovered and studied extensively over the past sixty years [Brs88,Deh00,GK03,Joy82,Kau91,Mat82,Taka42]. It appears that a special case of quandles (involutory quandles), called Kei, first appeared in literature in [Taka42], and studied for their algebraic structures in relation to symmetry of geometric objects. The name quandle is due to [Joy82].

The famous Fox tri-colorings (and $n$-colorings for $n\leq 3$, [Fox61]) of knot diagrams, that are identified with homomorphisms from the knot groups (the fundamental groups of the complements of knots) to dihedral groups, are generalized to colorings by quandles [FR92].

In [CJKLS03], quandle cohomology theory was constructed as a modification of rack cohomology theory [FRS95,FRS*], and quandle cocycle invariants were defined in a state-sum form. It is our goal in this chapter to review colorings of knot diagrams by quandles and quandle cocycle invariants.