Abstract
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.
Highlights
In this paper we introduce what can be thought of as a generalization of singular knots to what we call stuck knots
Let D be a diagram of an oriented stuck knot or link L with stuck crossings involving multiplicities in N, and the bracket of | D | denoted by h| D |i be characterized by the rules:
In modeling RNA folding diagrams we have considered a transformation T that inserts a negative stuck crossing and a positive usual crossing in place of a bond vertex with antiparallel strands
Summary
In this paper we introduce what can be thought of as a generalization of singular knots to what we call stuck knots. The stuck crossings are vertices which behave like the rigid vertices in singular knots, but with a convention of which strand passes over the other. A stuck knot is the equivalence class of stuck not diagrams subject to the Reidemeister-like moves given above. For a non-oriented stuck knot (one component link), the signed sticking number can be defined by the sum of signs of the stuck crossings obtained by any of the two possible orientations of the knot.
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