Take any link $L_1$ with an unknotted component $K$, cut along the disk bounded by K (which usually intersects some of the other components of $L_1$ transversally ), twist n times, and reglue. Let us denote by $L_2$ the resulting link.
It is easy to see that this operation of twisting may be described diagrammatically in the following manner : take a diagram $D_1$ of $L_1$, where the unknoted circle representing $K$ in $D_1$ is denoted $D_K$, take a disk which do not intersect $D_K$ and which contains the strands intersecting $D_K$ in $m$ points and replace that with a disk representing the diagram of an $m$-integer tangle which consists of $n$ twists to get a new diagram $D_2$.
My question is the following : if we start with two different diagrams $D_1$ and $D'_1$ of $L_1$ where $D_K$ and $D'_K$ are unknoted circles and perform the diagramatic twisting to the two diagrams to get $D_2$ and $D'_2$. Are these resulting diagrams equivalent (do they represent the same link $L_2$) ?
The question may be also put as : twisting two equivalent links along the same trivial component does it produce two equivalent links ?