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 °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 \( {H}_n^{\mathcal{K}}\left(A\left|q\right.\right) \) can be considered as linear forms on the n-th “power” of the knot \( \mathcal{K} \), 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 hep-th/0501235, that concrete slicing of the Universal Mandelbrot set is not essential for revealing its structure.
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Kononov, Y., Morozov, A. Colored HOMFLY and generalized Mandelbrot set. J. High Energ. Phys. 2015, 151 (2015). https://doi.org/10.1007/JHEP11(2015)151
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DOI: https://doi.org/10.1007/JHEP11(2015)151