Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Quandles are sets with an algebraic operation satisfying three axioms relating to the three Reidemeister moves of a classical knot diagram. Each arc, or broken surface component, of a knot diagram can be assigned an element of the quandle in such a way to satisfy the axioms of the operation. This assignment is called a quandle coloring of the knot.

Fox’s n-coloring, notably tri-coloring, is the coloring of a knot by Z/nZ. The number of colorings by a given quandle is an invariant; this invariant is commonly used to distinguished the trefoil from the unknot. Moreover, a (co)homology theory can be defined using quandles.

The quandle cocycle invariant has been used to determine the triple point number of surface-knots. A surface-knot has a singularity set consisting of double points, branch points and triple points. The triple point number of a surface-knot is the minimal number of triple points in any broken-surface diagram representing that knot.

I will talk about a result relating the triple-point number of surface-knots and the Reidemeister 3 number of classical knot diagrams.