SUMS: Gardiner -- Knot Theory

From the ancient Gordian knot to today’s inexplicably tangled headphone wires, people
have wanted to know whether knots are unknottable.  Considering for instance a knot tied
in a shoelace, a mathematician would argue that any knot-tying process is completely
reversible, and is therefore always unknottable.  But what if, after tying such a knot,
the two ends of the shoelace were fused together, and cutting wasn’t allowed? This is
essentially the topological definition of a knot, and the question of whether two
arbitrary knots are the same (or indeed unknottable) was a fundamental problem that knot
theorists tangled with up until the 20th century.  

Even today, the only known recognition algorithms are very slow and have unknown
complexity.  This talk will introduce the concept of topological knots and explore some
of the simpler approaches that were taken in attempts to prove the knot recognition
problem.  The content will be accessible to all audiences, and will work from basic
definitions to weak knot invariants to (time permitting) knot polynomials.