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arXiv · 1311.0526

Bounds on Übercrossing and Petal Numbers for Knots

Colin Adams·Orsola Capovilla-Searle·Jesse Freeman·Daniel Irvine·Samantha Petti·Daniel Vitek·Ashley Weber·Sicong Zhang
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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, $\text{ü}(K)$, 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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Colin Adams, Orsola Capovilla-Searle, Jesse Freeman, Daniel Irvine, Samantha Petti, Daniel Vitek, Ashley Weber, Sicong Zhang. 2013-11-03. Bounds on Übercrossing and Petal Numbers for Knots. https://arxiv.org/abs/1311.0526

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