The number of Cromwell moves needed for unknotting an arc-presentation of the trivial knot

点击翻译按钮获取中文摘要

A knot is a circle in the 3-dimensional space R3. Every knot can be placed in a figure in the form of an open-book (a book opened so that every adjacent pair of pages are tangent to each other only in the biding) so that it intersects every page in a single arc). We call such a placement of a knot an arc-presentation. A knot is called trivial if it lies in a plane after being moved continuously. The trivial knot has an arc-presentation with two arcs. An example of infinite sequence of arc-presentations with n arcs of the trivial knot as below is given. They need linearly many exchange moves not changing the number of arcs with respect to n until they admit a merge move decreasing the number of arcs.