abstract: From DNA experiments, it is known that type II topoisomerases can reduce the fraction of knots in DNA over that found in randomly cyclized DNA; the amount that the fraction of knots is reduced is one measure of ``knot reduction''. These topoisomerases act locally in the DNA by transiently breaking one strand of DNA to allow another strand to pass through (strand passage). Szafron and Soteros have been using a self-avoiding polygon model on the simple cubic lattice to model this strand passage action. The details of the combinatorial and topological theory behind this model will be reviewed. Then our recent Monte Carlo results for this model on how knot reduction depends on the local juxtaposition structure and the sign of the crossing at the strand passage site will be presented.