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).

  Polymer adsorbed onto a two dimensional surface:

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

    The Flat Figure-8- Theory predicts:

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

  Experiments on a granular chain twisted into a Figure-8 on a vibrating plane:

• 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

  DNA knots adsorbed onto a mica surface

• Ercoline, Valle, Adamcik, Witz, Metzler, Rios, Roca & Dietler, PRL 98, 058102 (2007).

    Dense Flat Knots: Polymers can become dense by application of pressure, or action of attractive forces.  Are tight shapes also present in topologically entangled dense polymers?

  Scaling analysis suggests that the trefoil is loose:

  Simulations by E. Orlandini, A.L. Stella, and C. Vanderzande, PRE68, 031804 (2003) confirm this:

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