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. 



Minimum number of double points over all immersed disks
For knots in $S^3=\partial B^4$ this is called the [four-ball crossing number or clasp number](http://www.indiana.edu/~knotinfo/descriptions/clasp_number.html).


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](http://www.indiana.edu/~knotinfo/).


