In this paper we study rational real algebraic knots in RP 3 . We show that two real rational algebraic knots of degree ≤ 5 are rigidly isotopic if and only if their degrees and encomplexed writhes are equal. We also show that any smooth irreducible knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree ≤ 6. Furthermore an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented. Date

In this paper we define and study flexible links and flexible isotopy in RP 3 ⊂ CP 3 . Flexible links are meant to capture the topological properties of real algebraic links. We classify all flexible links up to flexible isotopy using Ekholm's interpretation of Viro's encomplexed writhe.

We give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in P × R, where P is a punctured Riemann surface. As an application we show that, for any integer k and any homology class h ∈ H1(P × R), there are k Legendrian knots, all representing h, which are pairwise smoothly isotopic through a formal Legendrian isotopy, but which lie in mutually distinct Legendrian isotopy classes.

International audienceEquip each point x of a homogeneous Poisson point process P on R with Dx edge stubs, where the Dx are i.i.d. positive integer-valued random variables with distribution given by µ. Following the stable multi-matching scheme introduced by Deijfen, Häggström and Holroyd [1], we pair off edge stubs in a series of rounds to form the edge set of a graph G on the vertex set P. In this note, we answer questions of Deijfen, Holroyd and Peres [2] and Deijfen, Häggström and Holroyd [1] on percolation (the existence of an infinite connected component) in G. We prove that percolation may occur a.s. even if µ has support over odd integers. Furthermore, we show that for any ε > 0, there exists a distribution µ such that µ({1}) > 1 − ε, but percolation still occurs a.s.

Graph pebbling considers the problem of transforming configurations of discrete pebbles to certain target configurations on the vertices of a graph, using the so-called pebbling move. This paper provides counterexamples to a monotonicity conjecture stated by concerning the pebbling number compared to the pebbling threshold.