# Shortest Casson tower containing a slice disk for the attaching curve

A Casson tower is obtained as follows: Start with a properly immersed disk in $\mathbb{B}^4$ - a regular neighborhood of such a disk is called a kinky handle. The boundary of the core disk (necessarily in $\mathbb{S}^3$) is called the attaching circle. At each point of self-intersection of the core immersed disk, we have a simple closed curve by starting at the intersection point, leaving along one sheet and returning along the other. Call the collection of these curves (one for each self-intersection) the set of accessory circles. A Casson tower of height 1 is just a kinky handle. An Casson tower of height $n$ is obtained by attaching kinky handles to each of the accessory circles at the top stage of a Casson tower of height $n-1$. A Casson tower of infinite height is called a Casson handle and Freedman has shown that Casson handles are homeomorphic to $\mathbb{D}^2 \times \mathbb{D}^2$.

In 1982, Freedman showed that within a Casson tower of height 6 one can embed a Casson tower of height 7 with the same attaching circle, and as a result, Casson towers of height 6 contain Casson handles with the same attaching circle. In 1984, Gompf and Singh improved this by showing how to embed a height 6 tower inside a height 5 tower. I believe that there has been further improvement to this theorem. This paper (on pp 15) says that the improved number might be 4, and this one (on pp 103) appears to claim that it is 3. Unfortunately, neither of these above papers has a reference for these results.

With this background, I am seeking a reference for the answer to the following question:

What is the least integer $n$ for which it is known that given a Casson tower of height $n$, one can embed within it a Casson handle with the same attaching circle? Equivalently, what is the least integer $n$ such that given a Casson tower of height $n$, the attaching circle bounds an embedded topological disk within the given tower?

References:

1. M. Freedman, The topology of 4-dimensional manifolds, J. Differential Geom. 17 (1982), 357-453.
2. R. Gompf and S. Singh, On Freedman's reimbedding theorems, Four-Manifold Theory (C. Gordon and R. Kirby, eds.), Contemp. Math., vol. 35, Amer. Math. Soc., Providence, R.I., 1984, pp. 277-310.