A Convex Cover for Closed Unit Curves has Area at Least 0.0975

We combine geometric methods with a numerical box search algorithm to show that the minimal area of a convex set in the plane which can cover every closed plane curve of unit length is at least [Formula: see text]. This improves the best previous lower bound of [Formula: see text]. In fact, we show that the minimal area of the convex hull of circle, equilateral triangle, and rectangle of perimeter [Formula: see text] is between [Formula: see text] and [Formula: see text].

Minimal grid diagrams of 11 crossing prime alternating knots

The arc index of a knot is the minimal number of arcs in all arc presentations of the knot. An arc presentation of a knot can be shown in the form of a grid diagram which is a closed plane curve consisting of finitely many horizontal line segments and the same number of vertical line segments. The arc index of an alternating knot is its minimal crossing number plus two. In this paper, we give a list of minimal grid diagrams of the 11 crossing prime alternating knots obtained from arc presentations with 13 arcs.

Parity conditions for realizability of Gauss diagrams

We consider a problem of realizability of Gauss diagrams by closed plane curves where the plane curves have only double points of transversal self-intersection. We formulate the necessary and sufficient conditions for realizability. These conditions are based only on the parity of double and triple intersections of the chords in the Gauss diagram.