Paul Martin & Rob Sturman October 1, 2014
1 Introduction
Every child learns how to tie a knot, and some, like Scouts and Guides, even learn to distinguish between various types of knot. The classical mathematical field of knot theory uses ideas from topology and algebra to formally describe and hence mathematically model knots. In doing this it provides a scheme for classifying many different knots. This classification is a surprisingly hard problem, but the maths does an amazingly good job of it. A fundamental modern tool for modelling (in general) is the notion of entropy, a measure of complexity. (In the sense that a deck of cards is more complex than a set of poker dice, for example.) This project will explore this idea of complexity in the context of knots, and can take several different individual paths. Theoretically, group theory, graph theory, topology and abstract algebra intersect in the study of knots. In applied mathematics, the complexity of knots and braids can be harnessed in physical systems, such as fluid stirring devices. Computationally, there are many precise calculations to be made to visualise, compare and evaluate knots and their properties. This project will begin with a study of the basic mathematical definitions and properties of knots and braids, and then can lead in several different directions.
2 Knot and braid theory
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, considered up to continuous deformations. In knot theory, knots are therefore closed — the ‘ends’ of the string are joined together and cannot be undone — and non-self-intersecting. Some definitions might say that a knot cannot be untangled to give a simple loop (the unknot). It is already a difficult problem to determine whether a tangle of string can be manipulated into the unknot, or not. Knots, of course, are three-dimensional objects, but to apply mathematical theory they are typically con- sidered in two-dimensional projections such as in figure 1. They were originally classified by trial and error, but now they are most commonly cataloged based on the minimum number of crossings present.
Figure 1: Two knots, a trefoil and a cinquefoil, distinguishable by the number of crossings.
Knot theory typically asks questions such as: Can a given tangle of string be unknotted with cutting? When are two tangles deformable into each other? Can computational or algorithmic procedures be determined to help answer such questions?
1 One fundamental tool in deforming knots are the Reidermeister moves. These are local, simple operations which allow a knot to be manipulated. There are three types: I Twist or untwist a loop II Move one loop completely over another
III Move a thread completely over or under a crossing. These are more easily drawn than described, and can be seen in figure 2.
Figure 2: The three types of Reidermeister moves.
A closely related field is braid theory. Braids differ from knots in that they are not closed loops, but collections of twisted strings with fixed ends, such as in figure 3. (One question that could be considered in this project is: To what extent are knots and braids mathematically different?) Again, there are natural questions about when braids are equivalent; how to describe braids in terms of combinations of simple generators; what algebraic structures the generators possess; how to characterise the complexity of a braid.
Figure 3: A braid of three elements.
3 Braids, entropy and mixing
The question of complexity of a given knot or braid can be formalised in terms of entropy. One way to do so is to consider a dynamical process in which strands are braided using repeated simple moves - exactly as one would plait hair. In fact, plaiting hair is rather tame — the resulting braid is very ordered (in fact, zero entropy), but it’s fairly simple to achieve vast complexity by doing something very similar. This fact is exploited in fluid mixing applications, where periodic rearrangements of a small number of mixing ‘rods’ stir a fluid rapidly and effectively. An example is shown in figure 3. The theory of which braid protocols mix well, and which mix poorly, is a well-established result called the Thurston-Neilsen classification theorem. This theory lives in the intersection of topology, dynamical systems and geometry.
3.1 Bell-ringing and braids The English tradition of campanology, or bell-ringing, is an old one. Change-ringing dates from the fourteenth century. Church bells are typically not carefully tuned, and have prominent harmonics, so chordal music does Figure 4: Rapid mixing due to positive-entropy braiding of fluid stirrers. not sound good. Another constraint is that an individual bell cannot, for practical reasons of its size and weight, sound two or more notes in quick succession. For these reason, music for bells tends to be based on scales and arpeggios. In the English tradition, bells are rung sequentially, and the sequence is changed to create interesting patterns. Consider a numbering, in pitch order, of a collection of n bells. A row is a sequence in which each bell rings exactly once. The mathematical and musical interest in bell-ringing is in changing from one row to another. Here there is a crucial technical constraint: the weight and unwieldiness of the bells means that their chimes can only be delayed or advanced in time by a small amount. In practice this means that a particular bell in a sequence can only be moved one place at a time. That is, neighbouring bells may swap positions in a sequence, but no other pairs of bells. (A nice exercise in bell-ringing and combinatorics is to consider playing all possible rows, once and once only, in succession. This is known as ringing an extent.) Thus a particular sequence of rows defines a braid — physically, if you imagine the bell-ringers changing places whenever they change the order of ringing. An interesting question may be to compute the entropy of different extents.
4 Requirements
There are a number of modules of relevance to this project, including:
• MATH1920 (Computational Mathematics) • MATH2020 (Algebraic Structures 1) • MATH2025 (Algebraic Structures 2) • MATH2340 (The Mathematics of Music)
• MATH3071 (Groups and Symmetry) • MATH3225 (Topology) • MATH3396 (Dynamical Systems)
• MATH3397 (Nonlinear Dynamics) We expect that the students on this project have taken or intend to take at least one module from this list. Ideally the group of students taking this project will have a number of these modules covered between them. There may be opportunities for computational work (using Python, or possibly Matlab), but computation is not necessary for all participants.