TITLE: Minimum number of double points over all immersed disks
QUESTION [7 upvotes]: Let $K$ be a knot in the boundary of a compact smooth 4-manifold $X$, and suppose that $K$ is the the kernel of $\pi_1(\partial X) \to \pi_1(X)$.  Then $K$ is the boundary of some immersed disk $D \to X$ that has some fine number of double points.  I am interested in knowing how to compute minimum number of such double points, minimized over all such immersed disks.  For now, I will call it the immersion number of $K$ - denoted $I(K)$, for fun.  Has this quantity been studied in the literature?
One thing to note about $I(K)$ is that it provides a lower bound on the minimum genus of an orientable surface bounding $K$ (the "4-ball genus" if $X = B^4$).  I am interested in knowing examples of knots (even/especially with $X = B^4$) where $I(K)$ and the minimum genus of an orientable surface properly embedded in $X$ bounding $K$ are not equal.

REPLY [4 votes]: For knots in $S^3=\partial B^4$ this is called the four-ball crossing number or clasp number.
For knots in $S^3$ you have the following inequalities relating $I$ to the $4$-ball genus $g_4$ and the unknotting number $u$
$$g_4\leq I \leq u.$$
In general, these are no equalities. For counterexamples, you can have a look at the Table of Knot Invariants.