Knot Polynomials

By Liam Hernon, Monash University

Informally, a knot is what you get when you take a piece of string, tie it up, and glue the ends together. Many of the most complicated problems in knot theory stem directly from the seemingly innocuous concept of a knot invariant. A knot invariant is simply something which is the same for equivalent knots. They are ‘things’ which we can assign to a knot, and if two knots have different invariants, then we can say with confidence that those two knots are different. An example of a knot invariant is that of the stick number. The stick number is the minimum number of sticks required to create a knot, gluing the sticks end to end.

Knot polynomials are amongst the most common knot invariants, which incidentally lead to some very interesting questions. The central problem of my project is based on a very famous knot polynomial called the Jones polynomial. The Jones polynomial was discovered by Vaughan Jones in 1984 and has led to some very complicated problems in knot theory, one of which is the so-called slope conjecture and features exclusively throughout my report. Before describing the slope conjecture, it will be important to discuss a generalization of the Jones polynomial called the coloured Jones polynomials, and the concept of an essential surface. The coloured Jones polynomials are a sequence of Laurent polynomials for a knot which has been copied and pasted multiple times onto itself – that’s not quite the full or rigorous definition, but try and follow along. Loosely speaking, an essential surface of a knot is a surface which is bounded by the knot itself – it lives within the knot. Now that we’ve cleared up some terminology, let’s turn to the slope conjecture. The slope conjecture states that the maximal (or minimal) degree of each polynomial in the sequence of coloured Jones polynomials forms a quadratic, and this quadratic has a leading coefficient which is equal to the boundary slope of the essential surface.  It’s a bit of a mouthful, but really what the slope conjecture implies is that there exists a deep relationship between the algebraic and topological properties of a knot – something which is quite unexpected.

Liam Hernon was one of the recipients of a 2017/18 AMSI Vacation Research Scholarship.

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