Narrow Results

A cord is a simple curve on a punctured plane. We introduce diagrams which represent isotopy classes of cords. Using such diagrams, we prove that there is a one-to-one correspondence from the set of the isotopy classes of cords to a set of symmetric matrices whose components are non-negative integers, and we give necessary conditions for matrices to represent the isotopy classes of cords.

We prove that the modified von Koch snowflake curve, which we get as a limit by starting from an equilateral triangle (or from a segment) and repeatedly replacing the middle portion c of each interval by the other two sides of an equilateral triangle (and the corresponding von Koch snowflake domain), is non-self-intersecting if and only if c < ½. This answers a question of M. van den Berg.

We present an iterative method to define a two-parameter family of continuous functions f_a,θ: I → ℂ such that f_1/3,π/3 is the Koch curve. We consider the two-cases θ = π/3 and θ = π/4 of these generalized Koch curves f_a,θ(I). In each case we determine the pivotal value of a, the largest value of a for which the corresponding curve is not simple. We give characterizations of the double points of the curve (points on the curve that have two inverse images). In the case where θ = π/3 double points are vertices of equilateral triangles. When θ = π/4 the double points form Cantor sets in the plane.

We conclude with a more general result that proves that if the fixed set (attractor) of an iterated function system is connected, then it is a curve.

We present a two-parameter family of curves K_{a, θ} that are modifications of the Koch curve. For each θ, there is a corresponding pivotal valuea(θ) of a, where K_{a, θ} is simple if a > a(θ), and K_{a, θ} is self-intersecting if a < a(θ). We find a(θ), for θ ∈ (0, π/3], then show that the pivotal-valued curves comprise two classes of self-intersecting curves that we characterize by whether or not θ = π/n for some even positive integer n>2.