The average genus of a 2-bridge knot is asymptotically linear

TL;DR

This paper demonstrates that the average Seifert genus of 2-bridge knots increases linearly with crossing number, approaching a specific asymptotic formula, using a billiard table model.

Contribution

It introduces a billiard table model to analyze 2-bridge knots and establishes the asymptotic linear growth of their average genus.

Findings

Average genus approaches c/4 + 1/12 as crossing number c increases

Supports the linear growth hypothesis for knot genus

Provides a mathematical model for 2-bridge knot genus distribution

Abstract

Experimental work suggests that the Seifert genus of a knot grows linearly with respect to the crossing number of the knot. In this article, we use a billiard table model for $2$-bridge or rational knots to show that the average genus of a $2$-bridge knot with crossing number $c$ asymptotically approaches $c/4+1/12$.