Tutte seminar - David Jackson

A quantum invariant of knots

Abstract:

A (mathematical) knot is the everyday, perhaps mundane, object but with its two free ends having been smoothly sealed together. Two knots are to be regarded as 'equivalent' if one may be transformed into the other by the smooth operations of stretching or pulling. Thus, mathematically, a knot is an embedding of the unit circle into $\mathbb{R}^3$ considered up to ambient isotopy. 

A classical question is how to construct a function $\theta$ that distinguishes between knots. That is, for the set $\mathcal{K}$ of all knots, how to find a suitable set $\mathcal{S}$ and a function $\theta\colon\mathcal{K} \rightarrow\mathcal{S}$ such that $$\theta(\mathfrak{s}) \neq \theta(\mathfrak{t})\quad \Rightarrow \quad \mathfrak{s} \neq \mathfrak{t},$$ where $\mathfrak{s}$ and $\mathfrak{t}$ are knots. Such a function is called a \emph{knot invariant}. 
Not surprisingly, as a space embedding in another, knots and their higher dimensional analogues occur in an essential way in other branches of mathematics and in mathematical physics, as the appearance of the word "quantum" in the title suggests. 
I shall use Turaev's Theorem as the theme for this general talk. The idea is to project a knot into $\mathbb{R}^2$, equip the resulting image with over- and under-crossings to form a \emph{knot diagram} (so that the original knot may be reconstructed), confine all of the over- and under-crossings in the knot diagram to a \emph{braid diagram}, and then \emph{close} the braid diagram by joining the top and the bottom of the braid diagram by parallel strands. There are two immediate questions about this construction: "How may it be made bijective?" and "How does one use it to produce an actual invariant?" In this talk, I shall sketch the diagrammatic and algebraic ideas that lead to Turaev's Theorem, and thence to the existence of the Jones Polynomial. 
In a later talk, I shall discuss Vassiliev invariants and their connexions with combinatorial aspects of Hopf algebras and Lie algebras, and the association with homological ideas. 
On a purely personal note, my interest in knots was piqued when working on the toy model of string theory in the 1990s, using graph embeddings and matrix models, with Malcolm Perry at DAMTP in Cambridge and Terry Visentin at University of Winnipeg. Later, I was also enticed by ramified covers of the sphere and intersection questions in the moduli space of curves which, in spite of the perhaps unfamiliar terminology, contain a lot of combinatorics. The Goulden-Jackson-Vakil Conjecture on the existence of a particular moduli space is a topic of current interest in algebraic geometry.