Mathematical Sciences Department
Discrete Mathematics Seminar
Neranga Fernando, College of the Holy Cross
Monday, February 20, 2023
4:00 pm - 4:50 pm
Stratton Hall 202
Title: Quandles, racks and shelves"
Abstract: A quandle is a set X with a binary operation ∗ : X × X → X satisfying:
- For all x ∈ X, x ∗ x = x
- For all y ∈ X, the map βy : X → X defined by βy(x)= x ∗ y is invertible.
- For all x,y,z ∈ X, (x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z).
The three axioms of a quandle algebraically encode the three Reidemeister moves in knot theory.
A rack is a set with a binary operation that satisfies (ii) and (iii), and a shelf is a set with a binary operation satisfying self-distributivity (iii).
I will begin this talk by giving an introduction to quandles. I will then talk about racks, shelves and quandle rings. Moreover, I will talk about a recent joint work with Matthew Pradeep Goonewardena and Mohamed Elhamdadi on shelves.