We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in S 2 and in R 2 .
We study random knots and links in R 3 using the Petaluma model, which is based on the petal projections developed in . In this model we obtain a formula for the distribution of the linking number of a random two-component link. We also obtain formulas for the expectations and the higher moments of the Casson invariant and the order-3 knot invariant v3.… (More)
To any generic curve in an oriented surface there corresponds an oriented chord diagram, and any oriented chord diagram may be realized by a curve in some oriented surface. The genus of an oriented chord diagram is the minimal genus of an oriented surface in which it may be realized. Let g n denote the expected genus of a randomly chosen oriented chord… (More)