Introduction

A brief introduction to knots

History:
Art:
- Knots in art, by Piotr Pieranski (offline)
- Knot Art (web collection by Peter Suber)
Physics:
Polymers:
- Knots are topological invariants in ring (self-avoiding) polymers
- A sufficiently long polymer will almost certainly have knots:
  - H.L. Frisch & E. Wassermann (1961);
  - M. Delbrück (1962);
  - D.W. Summers & S.G. Whittington, J. Phys. A21, 1689 (1988).
- Knots are not included in the standard theory for self-avoiding polymers.
- Knots modify dynamics of polymers; e.g. relaxation or electrophoresis
Biology & Medicine:
- Knots in circular DNA and Topoisomerases
  - Lecture by A. Stasiak on DNA and Knots (offline)
  - Lifting the curtain: Using topology to probe the hidden action of enzymes (DeWitt Sumners)
- Tying a molecular knot with optical tweezers (in actin)
- Cancer therapy
Mathematics:
- Knot types are usually classified according to the number of crossing in their simplest planar projection.
  - Examples of `prime knots'

Identification of knots is difficult, because it is a global property that depends on the entire shape of the curve (need Knot invariants):
- J.W. Alexander (1923) polynomial: First algorithm which can distinguish between (some) knots
- Jones, HOMFLY, Kauffman polynomials, ... (KnotServer)
- "Ideal Knots," and "Knot Energies:" Can the knot type be determined from the `ideal' shape of the curve that minimizes a particular (two-body) potential? (ram video)