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Abstract: It is still unknown whether the Vassiliev invariants of knots classify all knots. It is known that the space of Vassiliev invariants of order m (mod those of lesser order) is isomorphic to the space of linear functionals on the space of chord diagrams of m chords (Kontsevich). A chord diagram is an oriented circle with pairs of points on the circle identified by joining them with chords. A chord diagram with its orientation reversed is the inverse of the original chord diagram. A chord diagram and its inverse are not necessarily the same. Certain very specific linear combinations of these chord diagrams are called Chinese characters.
We will examine some of the topological and combinatorial properties of Chinese characters, focusing on them as linear combinations of chord diagrams. The main result will be the decomposition of any Chinese character with an odd number of endpoints (univalent vertices) into a sum of chord diagrams minus their inverses. This calculation coincides with other results which imply that the search for a Vassiliev invariant which can discern non-invertible knots is a difficult one. An undergraduate background of linear algebra, a little topology and a little combinatorics is all that is required to follow the talk.