The shrinkage type of knots

Abstract

We study spectral gaps of cellular differentials for finite cyclic coverings of knot complements. Their asymptotics can be expressed in terms of irrationality exponents associated with ratios of logarithms of algebraic numbers determined by the first two Alexander polynomials. From this point of view, it is natural to subdivide all knots into three different types. We show that examples of all types abound and discuss what happens for twist and torus knots as well as knots with few crossings.