Bounds on übercrossing and petal numbers for knots

Abstract

An n-crossing is a point in the projection of a knot where n strands cross so that each strand bisects the crossing. An übercrossing projection has a single n-crossing and a petal projection has a single n-crossing such that there are no loops nested within others. The übercrossing number, ü(K), is the smallest n for which we can represent a knot K with a single n-crossing. The petal number is the number of loops in the minimal petal projection. In this paper, we relate the übercrossing number and petal number to well known invariants such as crossing number, bridge number, and unknotting number. We find that the bounds we have constructed are sharp for (r, r + 1)-torus knots. We also explore the behavior of übercrossing number under composition.