Classification of Alternating Knots with Tunnel Number One

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 [10]. Also, its genus ([3], [12]) and its crossing number ([7], [13], [17]) 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 S − int(N (K ∪ 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.