Average crosscap number of a 2-bridge knot

Moshe Cohen; Thomas Kindred; Adam M. Lowrance; Patrick D. Shanahan; Cornelia A. Van Cott

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arXiv · 2501.03099

Average crosscap number of a 2-bridge knot

Moshe Cohen·Thomas Kindred·Adam M. Lowrance·Patrick D. Shanahan·Cornelia A. Van Cott

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Abstract

We determine a simple condition on a particular state graph of an alternating knot or link diagram that characterizes when the unoriented genus and crosscap number coincide, extending work of Adams and Kindred. Building on this same work and using continued fraction expansions, we provide a new formula for the unoriented genus of a 2-bridge knot or link. We use recursion to obtain exact formulas for the average unoriented genus $\overlineΓ(c)$ and average crosscap number $\overlineγ(c)$ of all 2-bridge knots with crossing number $c$, and in particular we show that $\displaystyle{\lim_{c\to\infty} \left(\frac{c}{3}+\frac{1}{9} - \overlineΓ(c)\right) = \lim_{c\to\infty} \left(\frac{c}{3}+\frac{1}{9} - \overlineγ(c)\right) = 0}$.

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BibTeX RIS

Moshe Cohen, Thomas Kindred, Adam M. Lowrance, Patrick D. Shanahan, Cornelia A. Van Cott. 2025-08-06. Average crosscap number of a 2-bridge knot. https://arxiv.org/abs/2501.03099

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