Mathsyej875771187.fi Les.wordpress.com/20 18/07/3b1bexpo.pdf

Knots are a mathematical concept where a loop in space is tied in a way that cannot be untangled. Mathematicians care about knots because they are an incredibly hard puzzle, trying to determine if two knots can be transformed into each other. The puzzle is to find an efficient way to untie a given loop without cutting and regluing it. Knot theory is a field of math that deals with this problem, and it involves topology. The question being explored is whether a given knot can have a surface whose boundary is that knot. A surface can be created by giving a band or rectangle half-twists and gluing the ends together. The goal is to find a spanning surface for a given knot. Knots can be represented as knot diagrams, which are loops in a plane with double points where the loop passes through itself, and the over-under information is chosen at each double point. However, this procedure may produce a link instead of a knot. To ensure a knot, the diagram should represent a single loop. The unknotting algorithm involves switching crossings to untie a knot, and it starts by pinching the loop at a point and lifting it up, switching problematic crossings as needed. The algorithm can be described as choosing a point and direction on the knot, and switching crossings that have not been passed through before, to create a lifted loop that is a potentially twisted circle.