## Topological entanglement complexity of systems of polygons and walks in tubes

##### Abstract

In this thesis, motivated by modelling polymers, the topological entanglement complexity of systems of two self-avoiding polygons (2SAPs), stretched polygons and systems of self-avoiding walks (SSAWs) in a tubular sublattice of Z3 are investigated. In particular, knotting and linking probabilities are used to measure a polygon fs selfentanglement and its entanglement with other polygons respectively. For the case of 2SAPs, it is established that the homological linking probability goes to one at least as fast as 1-O(n^(-1/2)) and that the topological linking probability goes to one exponentially rapidly as n, the size of the 2SAP, goes to infinity. For the case of stretched polygons, used to model ring polymers under the influence of an external force f, it is shown that, no matter the strength or direction of the external force, the knotting probability goes to one exponentially as n, the size of the polygon, goes to infinity. Associating a two-component link to each stretched polygon, it is also proved that the topological linking probability goes to unity exponentially fast as n → ∞. Furthermore, a set of entangled chains confined to a tube is modelled by a system of self- and mutually avoiding walks (SSAW). It is shown that there exists a positive number γ such that the probability that an SSAW of size n has entanglement complexity (EC), as defined in this thesis, greater than γn approaches one exponentially as n → ∞. It is also established that EC of an SSAW is bounded above by a linear function of its size. Using a transfer-matrix approach, the asymptotic form of the free energy for the SSAW model is also obtained and the average edge-density for span m SSAWs is proved to approach a constant as m → ∞. Hence, it is shown that EC is a ggood h measure of entanglement complexity for dense polymer systems modelled by SSAWs, in particular, because EC increases linearly with system size, as the size of the system goes to infinity.

##### Degree

Doctor of Philosophy (Ph.D.)##### Department

Mathematics and Statistics##### Program

Mathematics and Statistics##### Supervisor

Soteros, Chris##### Committee

Kusalik, Anthony J. (Tony); Diao, Yuanan; Bremner, Murray R.; Bickis, Mikelis G.; Srinivasan, Raj##### Copyright Date

2008##### Subject

Tube

Polygon

Polymer

Self-avoiding walk

Entanglement complexity