AI 中文总结
本文提出一种多项式时间方法，从任意经典纽结图示构造规模为O(n^2)的局部极小桥表示，并指出平凡纽结识别属于P类。
AI 中文摘要
对于经典纽结，许多特征难以从纽结的任意图示中确定。桥数就属于此类特征。如Ozawa和Takao所示，一个纽结可能存在桥数仅为局部极小的桥表示。本文给出一种方法，从具有n个交点的任意经典纽结图示出发，构造该纽结的一个规模为O(n^2)的局部极小桥表示。该方法仅需多项式时间和空间。对于某些纽结类，任何局部极小值同时也是全局极小值，Otal已对平凡纽结证明了这一点。由于平凡纽结是唯一桥数为1的纽结，平凡纽结识别问题属于P类。
英文摘要
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 \(P\).
Comments21 pages, 12 figures, 4 tables. Comments are very welcome. Ancillary files are an AI-authored implementation of conversion to 3-page embedding sentence (not independently verified)
