Knots appear frequently in semiflexible (bio)polymers, including double-stranded DNA, and their presence can affect the polymer's physical and functional properties. In particular, it is possible and indeed often the case that multiple knots appear on a single chain, with effects that have been scrutinized only in the past few years.
Here we consider an idealization of a typical optical tweezers experiment involving a semiflexible double-knotted polymer, with steric hindrance and varying persistence length. Using exhaustive Molecular Dynamics simulations we show that the polymer persistence length influences the distribution of the entanglements; possibly more important, we observe and report how the relative chirality of the otherwise identical knots substantially modifies their interaction. We analyze the free energy of the chain and extract the effective interactions between embedded knots, rationalizing some of their pertinent features by means of simple effective models. We believe the salient aspect of the knot-knot interactions emerging from our study will be verified in a large number of semiflexible polymers under tension, with important consequences for the characterization and manipulation of these systems -- be they artificial or of biological origin -- and for their conceivable technological applications.