Introduction: Knots & Polymers

Alexander and the Gordian knot     Knots in art, by Piotr Pieranski (offline)          Knot Art (web collection by Peter Suber)

  Polymers:

• Knots are topological invariants in ring (self-avoiding) polymers,

  • and modify their dynamics; e.g. relaxation or electrophoresis.
• 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).
  • But knots are not included in the standard theory for self-avoiding polymers!

  Bio-polymers:

• Knots do occur in circular DNA,

  • Lecture by A. Stasiak on DNA and Knots (offline)
  • and actively manipulted by Topoisomerase enzymes
    • Lifting the curtain: Using topology to probe the hidden action of enzymes (DeWitt Sumners)
  • possibly involved in some oncogenes, suggesting a route to
    • Cancer therapy
• Some proteins have knotted structures
• Knots have been tied artificially by optical tweezers in actin

  Knot classification:

• Knot types are usually classified according to the number of crossing in their simplest planar projection.

  • Scottish mathematical Physics (Thomson, Maxwell, and Tait) (offline)
  • Examples of `prime knots'
  • .

  

  • P.G. Tait, Trans. Roy. Soc. Edinburgh 28, 145 (1876-7). (examples)
  • Example of  `composite knot'           31#31

    

• Identification of a knot 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)

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