Genera and crossing numbers of $2$-bridge knots

TL;DR

This paper calculates the average genus of 2-bridge knots for each crossing number and analyzes its asymptotic behavior as the crossing number increases.

Contribution

It provides the first explicit determination of the average genus of 2-bridge knots and describes its asymptotic trend.

Findings

Average genus of 2-bridge knots for fixed crossing number is determined.

Oblique asymptote of the average genus as crossing number grows is obtained.

Abstract

In this paper, we determine the average genus of all the $2$-bridge knots with a given crossing number. As a consequence, we obtain the oblique asymptote of this value as the crossing number grows.