We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or self-virtualizations. We provide a number of results which point out the differences between these various notions. The proofs are mainly based on the techniques of Gauss diagram formulae.
Benjamin AUDOUX. Paolo BELLINGERI. Jean-Baptiste MEILHAN. Emmanuel WAGNER. "On usual, virtual and welded knotted objects up to homotopy." J. Math. Soc. Japan 69 (3) 1079 - 1097, July, 2017. https://doi.org/10.2969/jmsj/06931079