Narrow Results

We present two physical systems which make Gauß sums experimentally accessible. The probability amplitude for a two-photon transition in an appropriate ladder system driven by a chirped laser pulse is determined by a Gauß sum. The autocorrelation function of a quantum rotor is also of the form of a Gauß sum. These examples suggest rules for determining prime factor components on the basis of the properties of Gauß sums. Moreover, we show how Gauß sums are related to the Riemann Zeta function.

We introduce a new approach to Vassiliev invariants. This approach deals with Vassiliev invariants directly on knots and does not make use of diagrams. We give a series of applications of this approach, (re)proving some new and known facts on Vassiliev invariants.

We study the new formulas of the first author for the degree-3-Vassiliev invariants for knots in the 3-sphere and solid torus and present some results obtained by them. We show that a knot with Jones polynomial consisting of exactly two monomials must have at least 20 crossings.

We give examples showing that the Fiedler solid torus degree 3 Gauß sum invariants can be used to detect mutation of links.