Bounds on \'{U}bercrossing and Petal Numbers for Knots
| Autor: | Adams, Colin, Capovilla-Searle, Orsola, Freeman, Jesse, Irvine, Daniel, Petti, Samantha, Vitek, Daniel, Weber, Ashley, Zhang, Sicong |
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| Rok vydání: | 2013 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | An $n$-crossing is a point in the projection of a knot where $n$ strands cross so that each strand bisects the crossing. An \"ubercrossing 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 \"ubercrossing number, $\text{\"u}(K)$, 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 \"{u}bercrossing 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 \"{u}bercrossing number under composition. Comment: 13 pages, 8 figures |
| Databáze: | arXiv |
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