Abstract

A book embedding of a complete graph is a spatial embedding whose planar projection has the vertices located along a circle, consecutive vertices are connected by arcs of the circle, and the projections of the remaining “interior” edges in the graph are straight line segments between the points on the circle representing the appropriate vertices. A random embedding of a complete graph can be generated by randomly assigning relative heights to these interior edges. We study a family of two-component links that arise as the realizations of pairs of disjoint cycles in these random embeddings of graphs. In particular, we show that the distribution of linking numbers of disjoint monotonic cycles can be described in terms of Eulerian numbers. Consequently, the mean of the squared linking number over all random embeddings is [Formula: see text], where [Formula: see text] is the number of interior edges in the cycles. We also show that the mean of the squared linking number over all pairs of disjoint monotonic [Formula: see text]-cycles in [Formula: see text] grows linearly in [Formula: see text].

Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call