Available student project - Knots, links and tangled nets

Research fields

  • Topological and Structural Science
  • Materials Science and Engineering
A simple tangled net. Can you find a knot? Why is it tangled?

Project details

Torus knots are well known to knot theoreticians: they are fomred by winding a simple loop in various ways around the surface of a torus. The same approach can be applied to the next simplest two-dimensional manifolds: a "bitorus" (genus 2) and "tritorus" (genus 3). Systematic tuning of the homotopy of single or multiple loops on these tori leads to knots and links in three-dimensional space. We will explore those knots and links by finding standard representations of them in 3d space using numerical "knot relaxation". The aim is to find simpler genus 2 and 3 knots and links. Advanced students will alo be able to investigate sinle tangled nets by a generalisation of this approach, mapping famililes of trees onto these tori rather than unbranched lines. The resulting patterns wil be multiple nets with varying degrees of entanglement. 

Required background

Some familiarity with elementary concepts of knot theory and two-dimensional geometry and topology would be good, though not essential. An interest in more complex aspects of molecular and framework material structures lies behind the project and interets in exploring forms via geoemrtic analysis is essential. 

Project suitability

This research project can be tailored to suit students of the following type(s)
  • 3rd year special project
  • PhB (2nd or 3rd year)
  • Honours project
  • Phd or Masters

Contact supervisor

Hyde, Stephen profile

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