Abstract
We consider self-avoiding polygons in a restricted geometry, namely an infinite L × M tube in Z3. These polygons are subjected to a force f, parallel to the infinite axis of the tube. When f > 0 the force stretches the polygons, while when f < 0 the force is compressive. In this extended abstract we obtain and prove the asymptotic form of the free energy in the limit f → −∞. We conjecture that the f → −∞ asymptote is the same as the free energy of Hamiltonian polygons, which visit every vertex in a L × M × N box.
Highlights
Self-avoiding walks and polygons are the standard lattice models of, respectively, linear and ring polymers in dilute solution (Vanderzande (1998))
Janse van Rensburg et al (2008) found that for sufficiently large fixed forces and sufficiently large polygons, all but exponentially few are knotted, i.e. the FWD-conjecture holds for sufficiently large forces
By restricting the polygons to lie in a lattice tube, Atapour et al (2009) proved that for any fixed force (either stretching (f > 0) or compressing (f < 0)), all but exponentially few sufficiently large polygons are knotted
Summary
Self-avoiding walks and polygons are the standard lattice models of, respectively, linear and ring polymers in dilute solution (Vanderzande (1998)). Regarding entanglement complexity, using a self-avoiding polygon model, Sumners and Whittington (1988) and independently Pippenger (1989) proved the 1960’s Frisch-Wasserman-Delbruck (FWD) conjecture that sufficiently long ring polymers will be knotted Janse van Rensburg et al (2008) found that for sufficiently large fixed forces and sufficiently large polygons, all but exponentially few are knotted, i.e. the FWD-conjecture holds for sufficiently large forces. By restricting the polygons to lie in a lattice tube, Atapour et al (2009) proved that for any fixed force (either stretching (f > 0) or compressing (f < 0)), all but exponentially few sufficiently large polygons are knotted (i.e. the FWD-conjecture holds).
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