Strict Inequalities for the $n$-crossing Number

Nicholas Hagedorn

doi:10.1142/s0218216523500281

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arXiv · 2212.12330

Strict Inequalities for the $n$-crossing Number

Nicholas Hagedorn

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Abstract

In 2013, Adams introduced the $n$-crossing number of a knot $K$, denoted by $c_n(K)$. Inequalities between the $2$-, $3$-, $4$-, and $5$-crossing numbers have been previously established. We prove $c_9(K)\leq c_3(K)-2$ for all knots $K$ that are not the trivial, trefoil, or figure-eight knot. We show this inequality is optimal and obtain previously unknown values of $c_9(K)$. We generalize this inequality to prove that $c_{13}(K) < c_{5}(K)$ for a certain set of classes of knots.

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Nicholas Hagedorn. 2022-11-26. Strict Inequalities for the $n$-crossing Number. https://doi.org/10.1142/s0218216523500281

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