A lower bound on the average genus of a 2-bridge knot

TL;DR

This paper establishes a lower bound on the average genus of 2-bridge knots by analyzing a truncated model using billiard table diagrams, providing insights into the growth of knot genus with crossing number.

Contribution

It introduces a new truncated model for 2-bridge knots and provides a method to estimate the average genus using Seifert circle counts.

Findings

Lower bound for average genus of 2-bridge knots derived

Model counts almost all knots twice

Method for counting Seifert circles in the model

Abstract

Experimental data from Dunfield et al using random grid diagrams suggests that the genus of a knot grows linearly with respect to the crossing number. Using billiard table diagrams of Chebyshev knots developed by Koseleff and Pecker and a random model of 2-bridge knots via these diagrams developed by the author with Krishnan and then with Even-Zohar and Krishnan, we introduce a further-truncated model of all 2-bridge knots of a given crossing number, almost all counted twice. We present a convenient way to count Seifert circles in this model and use this to compute a lower bound for the average Seifert genus of a 2-bridge knot of a given crossing number.