Average crosscap number of a 2-bridge knot

TL;DR

This paper provides a new formula for the unoriented genus of 2-bridge knots and links, characterizes when the unoriented genus and crosscap number are equal, and computes their average values as crossing number grows large.

Contribution

It introduces a simple condition on state graphs for alternating knots that links the unoriented genus and crosscap number, and derives formulas for their average values using continued fractions and recursion.

Findings

Derived a new formula for the unoriented genus of 2-bridge knots.

Established conditions when the unoriented genus equals the crosscap number.

Calculated the asymptotic behavior of average unoriented genus and crosscap number as crossing number increases.

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 $\overline{\Gamma}(c)$ and average crosscap number $\overline{\gamma}(c)$ of all 2-bridge knots with crossing number $c$, and in particular we show that $\displaystyle{\lim_{c\to\infty} \left(\frac{c}{3}+\frac{1}{9} - \overline{\Gamma}(c)\right) = \lim_{c\to\infty} \left(\frac{c}{3}+\frac{1}{9} - \overline{\gamma}(c)\right) = 0}$.