Stuck Knots

Singular knots and links have projections involving some usual crossings and some four-valent rigid vertices. Such vertices are symmetric in the sense that no strand overpasses the other. In this research we introduce stuck knots and links to represent physical knots and links with projections involving some stuck crossings, where the physical strands get stuck together showing which strand overpasses the other at a stuck crossing. We introduce the basic elements of the theory and we give some isotopy invariants of such knots including invariants which capture the chirality (mirror imaging) of such objects. We also introduce another natural class of stuck knots, which we call relatively stuck knots, where each stuck crossing has a stuckness factor that indicates to the value of stuckness at that crossing. Amazingly, a generalized version of Jones polynomial makes an invariant of such quantized knots and links. We give applications of stuck knots and links and their invariants in modeling and understanding bonded RNA foldings, and we explore the topology of such objects with invariants involving multiplicities at the bonds. Other perspectives are also discussed.