Thesis (PhD)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: In this dissertation we address the conservation of topological states in polymer knots.
Topological constraints are frequently included into theoretical descriptions of polymer
systems through invariants such as winding numbers and linking numbers of polynomial
invariants. In contrast, our approach is based on sequences of manipulations of knots that
maintain a given knot's topology; these are known as Reidemeister moves. We begin by
discussing basic properties of knots and their representations. In particular, we show how
the Reidemeister moves may be viewed as rules for dynamics of crossings in planar projections
of knots. Thereafter we consider various combinatoric enumeration procedures for
knot configurations that are equivalent under chosen topological constraints. Firstly, we
study a reduced system where only the zeroth and first Reidemeister moves are allowed, and
present a diagrammatic summation of all contributions to the associated partition function.
The partition function is then calculated under basic simplifying assumptions for the Boltzmann
weights associated with various configurations. Secondly, we present a combinatoric
scheme for enumerating all topologically equivalent configurations of a polymer strand that
is wound around a rod and closed. This system has the constraint of a fixed winding number,
which may be viewed in terms of manipulations that obey a Reidemeister move of the
second kind of the polymer relative to the rod. Again configurations are coupled to relevant
statistical weights, and the partition function is approximated. This result is used to calculate
various physical quantities for confined geometries. The work in that chapter is based
on a recent publication, "Conservation of polymer winding states: a combinatoric
approach", C.M. Rohwer, K.K. Müller-Nedebock, and F.-E. Mpiana Mulamba,
J. Phys. A: Math. Theor. 47 (2014) 065001. The remainder of the dissertation is
concerned with a dynamical description of the Reidemeister moves. We show how the rules
for crossing dynamics may be addressed in an operator formalism for stochastic dynamics.
Differential equations for densities and correlators for crossings on strands are calculated
for some of the Reidemeister moves. These quantities are shown to encode the relevant
dynamical constraints. Lastly we sketch some suggestions for the incorporation of themes
in this dissertation into an algorithm for the simulated annealing of knots. / AFRIKAANSE OPSOMMING: In hierdie tesis ondersoek ons die behoud van topologiese toestande in knope. Topologiese
dwangvoorwaardes word dikwels d.m.v. invariante soos windingsgetalle, skakelgetalle
en polinomiese invariante in die teoretiese beskrywings van polimere ingebou. In teenstelling
hiermee is ons benadering gebaseer op reekse knoopmanipulasies wat die topologie
van 'n gegewe knoop behou - die sogenaamde Reidemeisterskuiwe. Ons begin met 'n
bespreking van die basiese eienskappe van knope en hul daarstellings. Spesi ek toon ons
dat die Reidemeisterskuiwe beskryf kan word i.t.v. reëls vir die dinamika van kruisings
in planêre knoopprojeksies. Daarna beskou ons verskeie kombinatoriese prosedures om
ekwivalente knoopkon gurasies te genereer onderhewig aan gegewe topologiese dwangvoorwaardes.
Eerstens bestudeer ons 'n vereenvoudigde sisteem waar slegs die nulde en eerste
Reidemeisterskuiwe toegelaat word, en lei dan 'n diagrammatiese sommasie van alle bydraes
tot die geassosieerde toestandsfunksie af. Die partisiefunksie word dan bereken onderhewig
aan sekere vereenvoudigende aannames vir die Boltzmanngewigte wat met die verskeie kon-
gurasies geassosieer is. Tweedens stel ons 'n kombinatoriese skema voor om ekwivalente
kon gurasies te genereer vir 'n polimeer wat om 'n staaf gedraai word. Die beperking tot
'n vaste windingsgetal in hierdie sisteem kan daargestel word i.t.v. 'n Reidemeister skuif
van die polimeer t.o.v. die staaf. Weereens word kon gurasies gekoppel aan relevante
statistiese gewigte en die partisiefunksie word benader. Verskeie siese hoeveelhede word
dan bereken vir beperkte geometrie e. Die werk in di e hoofstuk is gebaseer op 'n onlangse
publikasie, "Conservation of polymer winding states: a combinatoric approach",
C.M. Rohwer, K.K. Müller-Nedebock, and F.-E. Mpiana Mulamba, J. Phys. A:
Math. Theor. 47 (2014) 065001. Die res van die tesis handel oor 'n dinamiese beskrywing
van die Reidemeisterskuiwe. Ons toon hoe die re els vir kruisingsdinamika beskryf kan
word i.t.v. 'n operatorformalisme vir stochastiese dinamika. Di erensiaalvergelykings vir
digthede en korrelatore vir kruisings op stringe word bereken vir sekere Reidemeisterskuiwe.
Daar word getoon dat hierdie hoeveelhede die relevante dinamiese beperkings respekteer.
Laastens maak ons 'n paar voorstelle vir hoe idees uit hierdie tesis geï nkorporeer kan word
in 'n algoritme vir die gesimuleerde vereenvoudiging van knope.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/86706 |
| Date | 04 1900 |
| Creators | Rohwer, Christian Matthias |
| Contributors | Muller-Nedebock, Kristian K., Scholtz, Frederik G., Stellenbosch University. Faculty of Science. Dept. of Physics. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | English |
| Type | Thesis |
| Format | xii, 145 p. |
| Rights | Stellenbosch University |
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