Knots and signal transmission in topological quantum devices

Abstract

We discuss the basic problem of signal transmission in quantum mechanics in terms of topological theories. Using the analogy between knot diagrams and quantum amplitudes we show how one can define and calculate the transmission coefficients of concept topological quantum devices that realize the knot topology. We observe that the problem is in different ways similar to that of transmission on quantum graphs. The desired transmission or filtering properties can be attained by the variation of topology of the device, or an external parameter, which in our model controls the topological phase. One interesting property of the transmission coefficients is the existence of ‘self-averaging’ phases, in which the value of the coefficient is independent from all the representatives in a chosen family of knots, and hence can be used for family classification. We briefly discuss physical realizations of the concept devices.