Communications in Analysis and Geometry
Volume 13 (2005)
Classification of Alternating Knots with Tunnel Number One
Pages: 151 – 186
An alternating diagram encodes a lot of information about a knot. For example, if an alternating knot is composite, this is evident from the diagram . Also, its genus (, ) and its crossing number (, , ) can be read off directly. In this paper, we apply this principle to alternating knots with tunnel number one. Recall that a knot K has tunnel number one if it has an unknotting tunnel, which is defined to be an arc t properly embedded in the knot exterior such that S3 - int(N (K union t)) is a handlebody. It is in general a very difficult problem to determine whether a given knot has tunnel number one, and if it has, to determine all its unknotting tunnels. In this paper, we give a complete classification of alternating knots with tunnel number one, and all their unknotting tunnels, up to an ambient isotopy of the knot exterior.