Roots of Alexander polynomials of random positive 3-braids

TL;DR

This work analyzes the roots of Alexander polynomials for knots and links that are closures of random positive 3-braids, revealing striking patterns and formulating precise conjectures. A central contribution is the introduction of a Lyapunov-exponent framework for the Burau representation, together with a bifurcation measure that conjecturally describes the limiting root distribution; partial results prove equidistribution on a region with positive mass and establish root-free zones near the origin. The authors prove significant structural results on unit-circle roots, including a two-thirds circle mass on the dominant arc and a central-limit theorem for the signature function, linking braid dynamics to spectral geometry. They also develop definiteness properties of the Burau representation, analyze real-axis and right-half-plane behavior, and demonstrate root-density phenomena on the unit circle, providing a rich, multi-faceted picture of how random positivity constraints shape polynomial roots in knot theory. The combination of experimental insight, rigorous spectral analysis, and probabilistic limit theorems offers a robust framework with potential to extend to larger braid groups and non-positive braids, with implications for knot invariants and dynamical systems in low-dimensional topology.

Abstract

Motivated by an observation of Dehornoy, we study the roots of Alexander polynomials of knots and links that are closures of positive 3-strand braids. We give experimental data on random such braids and find that the roots exhibit marked patterns, which we refine into precise conjectures. We then prove several results along those lines, for example that generically at least 69% of the roots are on the unit circle, which appears to be sharp. We also show there is a large root-free region near the origin. We further study the equidistribution properties of such roots by introducing a Lyapunov exponent of the Burau representation of random positive braids, and a corresponding bifurcation measure. In the spirit of Deroin and Dujardin, we conjecture that the bifurcation measure gives the limiting measure for such roots, and prove this on a region with positive limiting mass. We use tools including work of Gambaudo and Ghys on the signature function of links, for which we prove a central limit theorem.

Figures (13)

• Figure 1.0: Plots of the roots of the Alexander polynomials of two knots which are closures of 3-strand braids. In both cases, the polynomials have degree 764, but the one at left comes from a positive braid (of length 762) whereas the one at right is from an arbitrary braid (of length 1,598). The two braids were chosen randomly using the uniform measure on the monoid generators $\{\sigma_1, \sigma_2\}$ and $\{\sigma_1, \sigma_1^{-1}, \sigma_2, \sigma_2^{-1}\}$ respectively. For the positive braid at left, 69.3% of the roots are on the unit circle, including all 504 roots where $\mathop{\mathrm{Re}}\nolimits(z) > -0.5$, only four of which are roots of unity; this is compatible with Conjecture \ref{['conj:\n circle']}, which predicts asymptotically a value of 69.1%. In contrast, the arbitrary braid at right has 10.2% of its roots on the unit circle.
• Figure 1.0: The roots of $\Delta_K(t)$ for 2,500 knots $K$ coming from random positive 3-braids, chosen so the word lengths have mean 500 and standard deviation 170; some 1.2 million roots are plotted. The color of each point indicates the degree of $\Delta_K(t)$ where higher degrees are lighter colors.
• Figure 1.0: Another picture contrasting positive (top) versus generic (bottom) 3-braids, this time in the "trace coordinates" of Section \ref{['sec: alex props']}. Both plots are based on 2,500 random braid words chosen so that $\deg \Delta_K(t)$ has mean about 500 and standard deviation 170; the words in $\sigma_1, \sigma_2$ themselves have mean lengths of about 500 and 1,000 respectively. In the positive case, the words used are the same as in Figure \ref{['fig: pos many']}.
• Figure 1.0: A histogram of the roots on the upper half of the unit circle from Figure \ref{['fig: pos many']}, in terms of the usual polar angle $\theta$.
• Figure 1.0: To study what is happening in Figure \ref{['fig: pos many']} in the interior of the disc, consider those roots shown in the close-up at left. We radially project these onto the circular arc shown joining $-0.782$ to $\zeta_3$ (the choice of this arc is somewhat arbitrary). The resulting histogram is at right, where the horizontal axis is the angle in radians along said arc, measured clockwise from the real axis.

Theorems & Definitions (120)

• conjecture 1.1
• theorem 1.3
• theorem 1.4
• theorem 1.5
• theorem 1.7
• corollary 1.8
• theorem 1.10
• corollary 1.11
• lemma 2.2
• theorem 2.4: Burau1935