The k-fractal of a simplicial complex Original Research Article

The k-polynomial of a simplicial complex C is the function kC(x)=∑i⩾1 fixi where fi is the number of i-faces in C. These k-polynomials are closed under composition, and we are lead to ask: for higher composites of a complex C with itself, what happens to the roots of their k-polynomials? We prove that they converge to the Julia set of kC(x), thereby associating with C a fractal. For 2-dimensional complexes we exploit the Mandelbrot set to determine when their fractals are connected, and determine the connectness of the fractals for certain families of ‘stripped’ complexes.