Rational tangle distances on knots and links

In order to model biological reactions, distances between knots/links based on tangle replacement are defined. Given a tangle R , an R -move is defined as the replacement of the zero tangle in a link L with the tangle R . The R -distance between the link L and the link M is defined to be the minimum number of R -moves required to change L into M where the minimum is taken over all diagrams of the link L . A formula is given to determine when one 4-plat knot/link can be obtained from another 4-plat via one R -move when R is a rational tangle and not equal to the 1/ n tangle. The formula can also be used to find all rational tangles R for which the R -distance between two given 4-plats is 1 except in the case R = 1/ n tangle. These results are also generalized to the case when any rational tangle P is replaced with any rational tangle R , P not necessarily the zero tangle.