Average signature and 4-genus of 2-bridge knots

Abstract

We show that the average or expected absolute value of the signatures of all 2-bridge knots with crossing number $c$ approaches $\sqrt{2c/\pi }$. Baader, Kjuchukova, Lewark, Misev, and Ray consider a model for 2-bridge knot diagrams indexed by diagrammatic crossing number $n$ and show that the average 4-genus is sublinear in $n$. We build upon this result in two ways to obtain an upper bound for the average 4-genus of a 2-bridge knot: our model is indexed by crossing number $c$ and gives a specific sublinear upper bound of $9.75c/\log c$.

33 pages, 10 figures