Flat Knots

'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: Polymers can become dense by application of pressure, or action of attractive forces.

Are tight shapes also present in topologically entangled dense polymers?

Simulations and scaling analysis confirm that the flat figure-8 is remains tight:

Scaling analysis suggests that the trefoil is loose:

Simulations by E. Orlandini, A.L. Stella, and C. Vanderzande, PRE68, 031804 (2003) (cond-mat/0211259):

Delocalization in three dimensional compact polymers?

R.C. Lau, N.T. Moore, and A.Yu. Grosberg (cond-mat/0403413)
B. Marcone, E. Orlandini, A.L. Stella, and F. Zonta (cond-mat/040523)