Rational tangle distances on knots and links

Abstract

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.