The dilogarithm is a special function. Everybody knows about the logarithm. It is the function that turns multiplication into addition. It has a power series representation as


The dilogarithm has power series representation,


The dilogarithm is important because is shows up in two places. The first is in the computation of volume in hyperbolic geometry. Indeed, the volume of an ideal tetrahedron is expressible in terms of the dilogarithm. Also the Jones polynomial when computed in terms of special functions can be seen to be related to the dilogarithm.

The appearance of the dilogarithm in these seemingly unrelated places is one of the most provocative mathematical discoveries of the last few years. One of the most active areas of research in topology at the present time is the attempt to relate the Jones polynomial to the hyperbolic geometry of knots.

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