# Combinatorics and dynamics in polymer knots

Rohwer, Christian Matthias (2014-04)

Thesis (PhD)--Stellenbosch University, 2014.

Thesis

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.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/86706

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