Knot folding

In mathematical knot theory, knots are defined as a smooth embedding of topological circle into R3 and all its images [1, 3, 4, 6, 7]. These knots have no open ends, such that they cannot be tied or untied. If a knot can deform (an ambient isopoty) into a geometric circle, then it belongs to a special subgroup of knot, called unknot. For a mathematical knot, if we cut at finite locations, then the knot can be tied. These knots are the main subjects of this work, where we cut at only one location on a mathematical knot per topological circle. For a slight abuse of terminology, we call these knots physical knots. For simplicity, the rest of the paper will omit the word physical. However, instead of studying the phenomena of knots and how they hold together, which is the subject area of physical knot theory, we only study how to achieve a tightened knot from an untangled initial configuration. Physical knot theory is the study of mathematical models of knotting phenomena, often motivated by physical considerations from biology, chemistry, and physics [5].