A polynomial pair invariant of alternating knots and links

We introduce an invariant of alternating knots and links (called here WRP), namely a pair of integer polynomials associated with their two checkerboard planar graphs from their minimal diagram. We prove that the invariant is well-defined and give its values obtained from calculations for some knots in the tables. This invariant is strong enough to distinguish all knots in the tables with up to 10 crossings (including their mirror images)

Precarious citizenship: Rights claims of EU migrants in the UK

EU citizenship comprises a set of rights, but it is most closely associated with free movement across member states. However, while free movers formally enjoy equal civil, social, economic and residence rights throughout the European Union, informal application of these rights is spatially uneven. Further, political rights of EU citizens remain curtailed as free movers usually cannot vote in national elections in their country of residency, but they can vote in local and external elections. This dynamic of inclusion and exclusion generates a democratic deficit, but it also produces emergent spaces of political action. My thesis engages with this opening to analyse how EU migrant citizens claim their electoral, socioeconomic, and residence rights in Bristol: a culturally and economically vibrant English city, where the density of migrant networks generates a fertile ground for both political engagement and engaged research. The study is theoretically informed by geographic and political literatures on citizenship and interdisciplinary citizenship studies – with the acts of citizenship approach being particularly useful – as well as social and political theory. It is empirically grounded in the English context and relies on qualitative, place-based research in Bristol. This includes interviews and ethnography with migrant voters and political campaigners, and interviews with labour and community organisers. Data collection took place over an extended period before and after the EU referendum. The thesis distinguishes between the means and modes of political action. The means are defined as organised social practices that enable acts of citizenship, and include voting, organising, and campaigning. Three overarching modes of claiming European rights emerge through this analysis, and they include enactments along, across, and beyond national frames. In this way, personal and collective rights claims serve as empirical proxies, or windows onto, European migrant citizenship. The thesis argues that vulnerability is a powerful catalyst for political action unfolding through citizenship. It also shows that acts of citizenship, while often cast as revolutionary moments, are in fact processual and iterative. Citizenship understood in this way endows individuals with capacities and infrastructures to collectively learn, question, and rebel – to identify matters of concern, to identify sites of intensive relations of power, and to articulate interests and take action

EU Settlement Scheme: the cliff-edge approach puts many vulnerable applicants at risk

On 20 January 2020 Prime Minister Boris Johnson suffered his first defeat in Parliament. An amendment to the European Union (EU) Withdrawal Agreement Bill, moved by Liberal Democrat peer Jonny Oates with cross-party support, passed by 270 votes to 229. Lord Oates’ amendment wouldn’t break the EU Settlement Scheme, but it would fix it, argues Kuba Jabłonowski (University of Exeter)

Upper bound on delta-crossing and tabulation of knots up to four delta-crossings

We strengthened the known upper bound on the delta-crossing number of knots in therms of the triple-crossing number with additional assumptions. We also prove that we can always find at least one tangle from the set of four tangles, in any triple-crossing projections of any non-trivial knot or non-split link. In the last section, we enumerate and generate tables of minimal delta-diagrams for all prime knots up to the delta-crossing number equal to four.Comment: 9 page