Here is a knotted piece of rubber band:
Can you untangle this mess? In other words, can you move the band around until you get this:
For over a century, mathematicians have studied this and similar questions in knot theory.
A common strategy for attacking such a question is to use invariants. An invariant is simply a map $$ I\colon~ X \rightarrow Y $$ between sets \(X\) and \(Y\) such that \(I\) factors through the quotient \(X/\sim\) for some given equivalence relation \(\sim\) on \(X\). In our case, \(X\) is the set of all knots (i.e. embeddings of circles into 3-dimensional space) and we consider two knots to be equivalent under \(\sim\) if we can deform one into the other. \(Y\) can be anything: integers, functions, groups, sets, you name it!
In 1984, Vaughan Jones discovered a particularly fascinating knot invariant \(V\), which associates with each knot a Laurent polynomial. For example, for the first knot above—let's call it \(K\)— one computes $$ V_{K}(t)=-t^4 + t^3 + t $$ and for the second knot—let's call it \(U\) for unknot—we have $$ V_{U}(t)=1. $$
How does the Jones polynomial help us to decide if we can deform the knot \(K\) into \(U\)? Well, if we take for granted that it really is a knot invariant and that it takes those values on \(K\) and \(U\), we are already done! Indeed: Suppose \(K\sim U\). Then \(V_K(t)\) should be equal to \(V_U(t)\), since \(V\) is an invariant. But obviously, \(V_K\neq V_{U}\). So we cannot deform \(K\) into \(U\).
To make things more interesting, let's consider the following knot \(K'\):
It looks almost like our knot \(K\), but \(K'\not\sim K\). How do we see this, you ask? Well, we simply compute its Jones polynomial! $$ V_{K'}(t)=1 $$ Note that this is the same as for the unknot. However, we cannot deduce from this coincidence that \(K'\) can be deformed into the unknot! In fact, it is an open question whether there are any non-trivial knots with the same Jones polynomial as the unknot.
Exploring this and related questions is the main goal of this project.
Any textbook on knot theory that discusses Reidemeister moves, skein modules, and, of course, the Jones polynomial is a good place to start; see for instance [1] and [2]. After understanding the basic definitions and working out some examples, there are plenty of options open to you to explore: