Euler Characteristics of Knots.

The Euler characteristic of a knot in a closed 3-manifold is the maximal Euler characteristic among connected, orientable surfaces properly embedded in the exterior of the knot. The set of Euler characteristics of knots in S3 is {1,-1,-3,…}. The same is true for knots in the trivial homology class in any 3-manifold. In general, however, this is not the case for knots in a given non-trivial, finite-order homology class. We will discuss the behavior of the set of Euler characteristics of a homology class and pose some related questions.