Fox’s trapezoidal conjecture, proposed in 1962, states that the absolute values of the coefficients of the Alexander polynomial of an alternating knot form a trapezoidal sequence: they strictly increase, possibly plateau, and then strictly decrease. In this talk, I will discuss recent progress on the conjecture, based on joint work with András Juhász and Tamás Kálmán. In particular, we prove the conjecture for diagrammatic plumbings of special alternating links and obtain partial results for alternating three-braid closures

If you are curious about using your maths outside academia, want to learn new skills, or just want a change of pace from your PhD, then consider a policy internship. During a three-month UKRI policy internship at the Joint Nature Conservation Committee, I worked on assessing the impact of human-made underwater noise on harbour porpoises. I got to see what it was like to work for a government advisory body, and how scientific modelling is used to inform policy and real-world decision making, all whilst occasionally spotting dolphins from my office window. In this talk, I will describe my project and use it as a starting point to discuss internships more broadly: what you can gain from them, how they differ from academic research, and how to apply.