'Flat' Knots

Topologically constrained polymers in two dimensions, with a fixed number of crossings,

• A.Yu Grosberg and S.K. Nachaev, J. Phys. A25, 4659 (92).
• E. Guitter and E. Orlandini, J. Phys. A 32, 1359 (99).

Example 1: Polymer adsorbed onto a two dimensional surface:

• B. Maier and J.O. Radler, Phys. Rev. Lett. 82, 1911 (1999).

Example 2: 'Granular' chain on a vibrating plate

• E. Ben-Naim, Z.A. Daya, P. Vorobier, and R.E. Ecke, Phys. Rev. Lett. 86, 1414 (2001).

  The Flat Figure-8- Theory predicts:

Monte Carlo simulations of polymer flattened by a `gravitational field'

Vibrated Figure-8 experiment:

• M.B. Hastings, Z.A. Daya, E. Ben-Naim, and R.E. Ecke, Phys. Rev. Lett. E 66, 025102(R) (2002).

  The Flat Trefoil- Theory predicts a hierarchy of shapes (contractions)

Monte Carlo simulations of Trefoil:

  Swollen Flat Knots- All prime knots are tight; all composite knots are factored

  Dense Flat Knots: The figure-8 is tight, but the trefoil and other knots are loose [Dense Knots]

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