This talk on Quandles: The Algebra of Knots was held on Friday November 25, 2016 in MC 4020. The talk was given by Brennen Creighton-Young.
One of the major goals of knot theory is to determine whether or knot two knots can be continuously deformed into one another. This idea can be fully captured algebraically — and leads us to algebraic structures that not only capture the desired notion of knot deformations, but reveal themselves as truly fascinating mathematical objects. We will use only linear algebra and elementary abstract algebra.
Though continuous deformation is a straight forward idea, the rigorous definition of this, the notion of ambient isotopy, is practically unusable. This talk will provide a quick overview of basic knot theory and will work towards developing numerous algebraic strategies to identify when two knots are ambient isotopic. We will focus mostly on providing motivation for keis, an algebraic representation of knots, as well as their generalization, quandles. Quandles will prove to be not only helpful with respect to the goal of the classification of knots, but also as rich algebraic structures. The talk will involve small amounts of group theory, linear algebra and module theory.
Introduction to Knot Theory (15 mins)
Knots, links and knot diagrams
Knot equivalence and ambient isotopy
Knot invariants — crossing number, Fox tricoloring
Keis as Algebraic Representations of Knots (15 mins)
Definitions and motivation
The free kei on a set and the fundamental kei of a knot
Presentation matrix of a kei
Kei homomorphisms, colorings, the counting invariant
Quandles (15 mins)
The dihedral and Alexander quandles
Quandle homomorphisms and colorings
The Alexander modules and Alexander polynomial
In Conclusion (10 mins)
A brief discussion on generalizations
Knot theory and mathematical physics