Separability and the genus of a partial dual

Partial duality generalizes the fundamental concept of the geometric dual of an embedded graph. A partial dual is obtained by forming the geometric dual with respect to only a subset of edges. While geometric duality preserves the genus of an embedded graph, partial duality does not. Here we are interested in the problem of determining which edge sets of an embedded graph give rise to a partial dual of a given genus. This problem turns out to be intimately connected to the separability of the embedded graph. We determine how separability is related to the genus of a partial dual. We use this to characterize partial duals of graphs embedded in the plane, and in the real projective plane, in terms of a particular type of separation of an embedded graph. These characterizations are then used to determine a local move relating all partially dual graphs in the plane and in the real projective plane

A characterization of partially dual graphs

In this paper, we extend the recently introduced concept of partially dual ribbon graphs to graphs. We then go on to characterize partial duality of graphs in terms of bijections between edge sets of corresponding graphs. This result generalizes a well known result of J. Edmonds in which natural duality of graphs is characterized in terms of edge correspondence, and gives a combinatorial characterization of partial duality.Comment: V2: the statement of the main result has been changed. To appear in JGT

Integration and conjugacy in knot theory

This thesis consists of three self-contained chapters. The first two concern quantum invariants of links and three manifolds and the third contains results on the word problem for link groups. In chapter 1 we relate the tree part of the Aarhus integral to the mu-invariants of string-links in homology balls thus generalizing results of Habegger and Masbaum. There is a folklore result in physics saying that the Feynman integration of an exponential is itself an exponential. In chapter 2 we state and prove an exact formulation of this statement in the language which is used in the theory of finite type invariants. The final chapter is concerned with properties of link groups. In particular we study the relationship between known solutions from small cancellation theory and normal surface theory for the word and conjugacy problems of the groups of (prime) alternating links. We show that two of the algorithms in the literature for solving the word problem, each using one of the two approaches, are the same. Then, by considering small cancellation methods, we give a normal surface solution to the conjugacy problem of these link groups and characterize the conjugacy classes. Finally as an application of the small cancellation properties of link groups we give a new proof that alternating links are non-trivial.Comment: University of Warwick Ph.D. thesi

NSW recorded crime statistics 2012

  This report finds that in New South Wales, the surge in incidents involving the discharge of a firearm into premises which began in February 2010 has been reversed. In February 2010, incidents of discharge firearm into premises were running at an average of five a month. They peaked in August 2012 at an average rate of more than 11 per month. By December 2012, they were back down to around 6.7 a month. Other types of shooting incident are also down from their peaks between 2001 and 2003. This report presents further detail on crime trends across New South Wales. It finds that between January 2011 and December 2012: Fraud increased by 14.6 per cent; Assault - Non-domestic related fell by 5.7 per cent; Break and enter (non-dwelling) fell by 4.9 per cent; and Motor vehicle theft fell by 7.0 per cent The remaining major categories of crime remained stable across the state as a whole. &nbsp

Inaccurate approximations in the modeling of hyper-inflations

In time series macroeconometric models, the first difference in the logarithm of a variable is routinely used to represent the rate of change of that variable. It is often overlooked that the assumed approximation is accurate only if the rates of change are small. Models of hyper-inflation are a case in point, since in these models, by definition, changes in price are large. In this letter, Cagan's model is applied to Hungarian hyper-inflation data. It is then demonstrated that use of the approximation in the formation of the price inflation variable is causing an upward bias in the model's key parameter, and therefore an exaggeration of the effect postulated by Cagan.

Partial duality and Bollobas and Riordan's ribbon graph polynomial

Recently S. Chmutov introduced a generalization of the dual of a ribbon (or embedded) graph and proved a relation between Bollobas and Riordan's ribbon graph polynomial of a ribbon graph and its generalized duals. Here I show that the duality relation satisfied by the ribbon graph polynomial can be understood in terms of knot theory and I give a simple proof of the relation via the homfly polynomial of a knot

Magnetic helicity fluxes and their effect on stellar dynamos

Magnetic helicity fluxes in turbulently driven alpha^2 dynamos are studied to demonstrate their ability to alleviate catastrophic quenching. A one-dimensional mean-field formalism is used to achieve magnetic Reynolds numbers of the order of 10^5. We study both diffusive magnetic helicity fluxes through the mid-plane as well as those resulting from the recently proposed alternate dynamic quenching formalism. By adding shear we make a parameter scan for the critical values of the shear and forcing parameters for which dynamo action occurs. For this $\alpha\Omega$ dynamo we find that the preferred mode is antisymmetric about the mid-plane. This is also verified in 3-D direct numerical simulations.Comment: 5 pages, 6 figures, proceedings of IAU Symp. 286, Comparative Magnetic Minima: characterizing quiet times in the Sun and star