No abstract

Let D be a knot diagram, and let D denote the set of diagrams that can be obtained from D by crossing exchanges. If D has n crossings, then D consists of 2 n diagrams. A folklore argument shows that at least one of these 2 n diagrams is unknot, from which it follows that every diagram has finite unknotting number. It is easy to see that this argument can be used to show that actually D has more than one unknot diagram, but it cannot yield more than 4n unknot diagrams. We improve this linear bound to a superpolynomial bound, by showing that at least 2 3 √ n of the diagrams in D are unknot. We also show that either all the diagrams in D are unknot, or there is a diagram in D that is a diagram of the trefoil knot. arXiv:1710.06470v1 [math.CO]

J.-P. Roudneff has conjectured that every arrangement of $n\ge 2d+1\ge 5$ (pseudo) hyperplanes in the real projective space $\mathbb{P}^d$ has at most $\sum_{i=0}^{d-2} \binom{n-1}{i}$ cells bounded by each hyperplane. In this note, we show the validity of this conjecture for arrangements arising from Lawrence oriented matroids.

It is known that cyclic arrangements are the only unavoidable simple arrangements of pseudolines: for each fixed m ≥ 1, every sufficiently large simple arrangement of pseudolines has a cyclic subarrangement of size m. In the same spirit, we show that there are three unavoidable arrangements of pseudocircles.