Locally Minimal Bridge Presentations of Knots

RESUMO

For classical knots, many characteristics are difficult to determine from arbitrary diagrams of the knot. Bridge number is of this type. It is possible to have a bridge presentation of a knot in which the number of bridges is only locally minimal, as shown by Ozawa and Takao. Here, we give a method for beginning with an arbitrary classical knot diagram having \(n\) crossings and producing a locally minimal bridge presentation of the knot with size \(O(n^2)\). This method requires only polynomial time and space. For some classes of knots, any local minimum is also the global minimum, which was shown for the trivial knot by Otal. Because the unknot is the only knot with bridge number \(1\), unknot recognition is in \(P\).