Particles dispersed in a nematic yield complex defect structures. Beside structures where defects are localized close to particles also structures where disclination lines entangle two or more particles are found. Here we focus on dimmers and show that a simple geometrical description based on local differences can be used to classify configurations and describe restructurings among possible forms. The linking number of a disclination loop, which tells how the director field of a defect line twists when following a disclination, help us to distinguish between topologically different classes. Simple ring defects have zero linking number while entangled states exhibit fractional linking numbers.