A central limit theorem for the signatures of 2-bridge knots

TL;DR

This paper establishes a central limit theorem for the signatures of 2-bridge knots, showing their distribution approaches normality as crossing number increases, and provides a closed formula for counting knots with given signature.

Contribution

It derives a closed formula for the number of 2-bridge knots with specific signatures and proves the distribution of their signatures converges to a normal distribution.

Findings

Distribution of signatures approaches normal as crossing number increases

Closed formula for counting knots with given signature

Signature statistics computed for all 2-bridge knots with crossing number c

Abstract

Cohen, Lowrance, Madras, and Raanes computed the average (absolute value of) signature over all 2-bridge knots with crossing number $c$ by introducing the number $s(c,\sigma)$ of 2-bridge knots of crossing number $c$ and signature $\sigma$. Here we provide a closed formula for this number. We use these calculations to show that the distribution of the signatures of 2-bridge knots with crossing number $c$ approaches a normal distribution as $c$ tends to infinity.