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Abstract We prove the meridional rank conjecture for twisted links and arborescent links associated to bipartite trees with even weights. These links are substantial generalizations of pretzels and two-bridge links, respectively. Lower bounds on meridional rank are obtained via Coxeter quotients of the groups of link complements. Matching upper bounds on bridge number are found using the Wirtinger numbers of link diagrams, a combinatorial tool developed by the authors. 

Abstract We use graph theory simulations and single molecule experiments to investigate percolation properties of kinetoplasts, the topologically linked mitochondrial DNA from trypanosome parasites. The edges of some kinetoplast networks contain a fiber of redundantly catenated DNA loops, but previous investigations of kinetoplast topology did not take this into account. Our graph simulations track the size of connected components in lattices as nodes are removed, analogous to the removal of minicircles from kinetoplasts. We find that when the edge loop is taken into account, the largest component after the network de‐percolates is a remnant of the edge loop, before it undergoes a second percolation transition and breaks apart. This implies that stochastically removing minicircles from kinetoplast DNA would isolate large polycatenanes, which is observed in experiments that use photonicking to stochastically destroy kinetoplasts fromCrithidia fasciculata. Our results imply kinetoplasts may be used as a source of linear polycatenanes for future experiments. 

We prove that the knots [Formula: see text] and [Formula: see text] both have stick number 10. These are the first non-torus prime knots with more than 9 crossings for which the exact stick number is known. 

In this paper, we use 3-manifold techniques to illuminate the structure of the category of tangles. In particular, we show that every idempotent morphism [Formula: see text] in such a category naturally splits as [Formula: see text] such that [Formula: see text] is an identity morphism.