Braid variants and their applications

In Part I we develop the theory of arcbraids and arclinks, which are generalisations of the usual notions of braids and links; an alternative name for arclinks is irrational tangles. A cubical set without degeneracies is called a D-set. Just as braids induce rack automorphisms, arcbraids induce rack homomorphisms. We show that the formulae of the homomorphism induced by certain arcbraids is identical to that of the face maps of □-sets. Thus we can model the face maps of a O-set by arcbraids. However, there are many other arcbraids that do not model the usual face maps. We give a method for constructing new Q-sets, with unusual face maps, from arcbraids. Using this method, we construct three Q-sets. An alternating sum of the face maps of a □-set is the boundary operator of the chain complex associated to the classifying space of the D-set. So, in theory, new formulae for face maps could give rise to new homology theories. We show that quasi O-maps, a generalisation of □-maps, induce homeomorphisms of the corresponding classifying spaces. Furthermore, we show that we can form quasi Q-maps between the three O-sets constructed. Unfortunately, this confounds the hope for new homology theories, but only in this case! In Part IIwe define the Welded Jones polynomial, which is a nontrivial, welded isotopy invariant of welded links. In Chapter 5, signed Gauss codes are related to the fundamental rack; we give algorithms to compute the effects of operations such as reversing, mirroring, crossing changing and smoothing on these objects. We recall that a signed Gauss code corresponds to a virtual link. In Chapter 6 we show that permuting consecutive o’s in the code is equivalent to the extra isotopy move required for welded links. This allows us to define the Welded Bracket polynomial, which is actually a quotient of the Bracket polynomial of virtual links, and the Welded Jones polynomial can be obtained from this. We give nontrivial examples of computations which distinguish welded links. A theorem of Jones for classical knots, which does not hold for virtual or welded knots, implies that the Welded Jones polynomial is trivial for classical knots. A slight modification leads to the Welded W-polynomial, which is a nontrivial, welded isotopy invariant of classical knots. We end on the entertaining note that whereas the Jones polynomial of the connected sum of classical knots is the product of the individual polynomials, for the Welded W-polynomial it is the sum of the individual polynomials

Queerying activism through the lens of the sociology of everyday life

The approaching 30th anniversary of the introduction of the 1988 Local Government Act offers an opportunity to reflect on the nature of lesbian, gay and bisexual (LGB) activism in Britain. The protests against its implementation involved some of the most iconic moments of queer activism. Important though they are, these singular, totemic moments give rise to, and are sustained by small, almost unobtrusive acts which form part of LGB people’s everyday lives. This article aims to contribute to a re-thinking of queer activism where iconic activism is placed in a synergetic relationship with the quieter practices in the quotidian lives of LGB people. The authors interrogate a series of examples, drawn from three studies, to expand ideas about how activism is constituted in everyday life. They discuss the findings in relation to three themes: the need to forge social bonds often forms a prompt to action; disrupting the binary dualism between making history and making a life; and the transformative potential of everyday actions/activism. The lens of the sociology of everyday life (1) encourages a wider constituency of others to engage in politics, and (2) problematises the place of iconic activism.Peer reviewe

Type Annotation for Adaptive Systems

We introduce type annotations as a flexible typing mechanism for graph systems and discuss their advantages with respect to classical typing based on graph morphisms. In this approach the type system is incorporated with the graph and elements can adapt to changes in context by changing their type annotations. We discuss some case studies in which this mechanism is relevant.Comment: In Proceedings GaM 2016, arXiv:1612.0105

Euler Diagram Transformations

Euler diagrams are a visual language which are used for purposes such as the presentation of set-based data or as the basis of visual logical languages which can be utilised for software specification and reasoning. Such Euler diagram reasoning systems tend to be defined at an abstract level, and the concrete level is simply a visualisation of an abstract model, thereby capturing some subset of the usual boolean logic. The visualisation process tends to be divorced from the data transformation process thereby affecting the user's mental map and reducing the effectiveness of the diagrammatic notation. Furthermore, geometric and topological constraints, called wellformedness conditions, are often placed on the concrete diagrams to try to reduce human comprehension errors, and the effects of these conditions are not modelled in these systems. We view Euler diagrams as a type of graph, where the faces that are present are the key features that convey information and we provide transformations at the dual graph level that correspond to transformations of Euler diagrams, both in terms of editing moves and logical reasoning moves. This original approach gives a correspondence between manipulations of diagrams at an abstract level (such as logical reasoning steps, or simply an update of information) and the manipulation at a concrete level. Thus we facilitate the presentation of diagram changes in a manner that preserves the mental map. The approach will facilitate the realisation of reasoning systems at the concrete level; this has the potential to provide diagrammatic reasoning systems that are inherently different from symbolic logics due to natural geometric constraints. We provide a particular concrete transformation system which preserves the important criteria of planarity and connectivity, which may form part of a framework encompassing multiple concrete systems each adhering to different sets of wellformedness conditions

Andersen-Tawil Syndrome

Andersen-Tawil syndrome (ATS) is a rare condition consisting of ventricular arrhythmias, periodic paralysis, and dysmorphic features. In 2001, mutations in KCNJ2, which encodes the α subunit of the potassium channel Kir2.1, were identified in patients with ATS. To date, KCNJ2 is the only gene implicated in ATS, accounting for approximately 60% of cases. ATS is a unique channelopathy, and represents the first link between cardiac and skeletal muscle excitability. The arrhythmias observed in ATS are distinctive; patients may be asymptomatic, or minimally symptomatic despite a high arrhythmia burden with frequent ventricular ectopy and bidirectional ventricular tachycardia. However, patients remain at risk for life-threatening arrhythmias, including torsades de pointes and ventricular fibrillation, albeit less commonly than observed in other genetic arrhythmia syndromes. The characteristic heterogeneity at both the genotypic and phenotypic levels contribute to the continued difficulties with appropriate diagnosis, risk stratification, and effective therapy. The initial recognition of a syndromic association of clinically diverse symptoms, and the subsequent identification of the underlying molecular genetic basis of ATS has enhanced both clinical care, and our understanding of the critical function of Kir2.1 on skeletal muscle excitability and cardiac action potentia

A New Language for the Visualization of Logic and reasoning

Many visual languages based on Euler diagrams have emerged for expressing relationships between sets. The expressive power of these languages varies, but the majority can only express statements involving unary relations and, sometimes, equality. We present a new visual language called Visual First Order Logic (VFOL) that was developed from work on constraint diagrams which are designed for software specification. VFOL is likely to be useful for software specification, because it is similar to constraint diagrams, and may also fit into a Z-like framework. We show that for every First Order Predicate Logic (FOPL) formula there exists a semantically equivalent VFOL diagram. The translation we give from FOPL to VFOL is natural and, as such, VFOL could also be used to teach FOPL, for example