Some instances where topological illustration induced new mathematics

We shall present instances from generalized knot theory, braid theory and their interactions, where illustration promoted understanding and inspired new mathematics. The first instance addresses a question of V.F.R. Jones whether one can make analogous constructions to the (2-variable) Jones polynomial using other braid groups and other types of Hecke algebras. The second instance addresses the question of formulating braid equivalences, analogous to the Markov theorem for classical braids, in settings where we may not even have available algebraic structures for the related braids. The third instance is about the theory of bonded knots and bonded knotoids used for modelling proteins.