Obtaining graph knots by twisting unknots

Let K be a knot in the 3-sphere S3 and D a disk in S3 meeting K transversely more than once in the interior. For nontriviality we assume that |D∩K|⩾2 over all isotopies of K in S3−∂D. Let KD,n (⊂S3) be a knot obtained from K by n twisting along the disk D. We prove that if K is a trivial knot and KD,n is a graph knot, then |n|⩽1 or K and D form a special pair which we call an “exceptional pair”. As a corollary, if (K,D) is not an exceptional pair, then by twisting unknot K more than once (in the positive or the negative direction) along the disk D, we always obtain a knot with positive Gromov volume. We will also show that there are infinitely many graph knots each of which is obtained from a trivial knot by twisting, but its companion knot cannot be obtained in such a manner.