Bounds on Übercrossing and Petal Numbers for Knots

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, $\text{ü}(K)$, 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 tight for $(r, r+1)$-torus knots. We also explore the behavior of übercrossing number under composition.

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