Ma–Qiu index, presentation distance, and local moves in knot theory

The Ma–Qiu index of a group is the minimum number of normal generators of the commutator subgroup. We show that the Ma–Qiu index gives a lower bound of the presentation distance of two groups, the minimum number of relator replacements to change one group to the other. Since many local moves in knot theory induce relator replacements in knot groups, this shows that the Ma–Qiu index of knot groups gives a lower bound of the Gordian distance based on various local moves. In particular, this gives a unified and simple proof of the Nakanishi index bounds of various unknotting numbers, including virtual or welded knot cases.