Colored HOMFLY and Generalized Mandelbrot set

TL;DR

This paper explores the Mandelbrot property in the context of colored HOMFLY polynomials and their resultants, revealing Mandelbrot-like structures in the complex plane that relate to knot theory and iterated maps.

Contribution

It introduces the concept of Mandelbrot property for resultant-zeroes patterns of colored HOMFLY polynomials, extending the Mandelbrot set analogy to knot invariants.

Findings

Resultant-zeroes form Mandelbrot-like patterns in the complex A-plane.

Mandelbrot structures are observed despite A not being an ideal moduli parameter.

The patterns support the idea that Mandelbrot set features are visible even in complex knot invariants.

Abstract

Mandelbrot set is a closure of the set of zeroes of $resultant_x(F_n,F_m)$ for iterated maps $F_n(x)=f^{\circ n}(x)-x$ in the moduli space of maps $f(x)$. The wonderful fact is that for a given $n$ all zeroes are not chaotically scattered around the moduli space, but lie on smooth curves, with just a few cusps, located at zeroes of $discriminant_x(F_n)$. We call this phenomenon the Mandelbrot property. If approached by the cabling method, symmetrically-colored HOMFLY polynomials $H^{\cal K}_n(A|q)$ can be considered as linear forms on the $n$-th "power" of the knot ${\cal K}$, and one can wonder if zeroes of $resultant_{q^2}(H_n,H_m)$ can also possess the Mandelbrot property. We present and discuss such resultant-zeroes patterns in the complex-$A$ plane. Though $A$ is hardly an adequate parameter to describe the moduli space of knots, the Mandelbrot-like structure is clearly seen -- in full accord with the vision of arXiv:hep-th/0501235, that concrete slicing of the Universal Mandelbrot set is not essential for revealing its structure.