论文标题
特殊交流结的ropelength
The Ropelength of Special Alternating Knots
论文作者
论文摘要
长期以来的开放猜想指出，如果链接$ \ MATHCAL {K} $交替，则其RopeLength $ l（\ Mathcal {K}）$至少是顺序$ O（cr（\ Mathcal {K}））$。最近的结果表明，链接的最大编织索引从下面界定了链接的ropelgength。因此，在这种情况下，交替的链接具有与其最小交叉数成正比的最大辫子索引，例如$ t（2,2n）$ torus链接，然后该链接的ropelength在下面的最小交叉数的常数倍数。但是，如果链接的最大辫子索引与其交叉数相比很小，那么关于其ropemength是否以其交叉数的常数倍数界定的结果尚无已知结果。例如，$ t（2,2n+1）$ torus结的最小结图看起来与$ t（2,2n）$ torus链接几乎相同，但是它的ropelength是否在下面的$ n $ of $ n $的频率限制为迄今为止。在本文中，我们提供了第一个这样的结果，实际上是大量的交替结。具体而言，我们证明，如果交替结（即与一个组件的链接）具有降低的交替结图，在该图中，交叉的结节是正面或全部负数的（这样的结为特殊的交替结），则其ropeLength的ropeentenge是由其交叉数的常数倍数界定的。
A long standing open conjecture states that if a link $\mathcal{K}$ is alternating, then its ropelength $L(\mathcal{K})$ is at least of the order $O(Cr(\mathcal{K}))$. A recent result shows that the maximum braid index of a link bounds the ropelength of the link from below. Thus in the case an alternating link has a maximum braid index proportional to its minimum crossing number, such as the $T(2,2n)$ torus link, then the ropelength of the link is bounded below by a constant multiple of its minimum crossing number. However if the maximum braid index of a link is small compared to its crossing number, then there are no known results about whether its ropelength is bounded below by a constant multiple of its crossing number. For example, the $T(2,2n+1)$ torus knot has a minimum knot diagram that looks almost identical to that of the $T(2,2n)$ torus link, yet whether its ropelength is bounded below by a constant multiple of $n$ remains open to date. In this paper, we provide a first such result, and in fact for a large class of alternating knots. Specifically, we prove that if an alternating knot (namely a link with only one component) has a reduced alternating knot diagram in which the crossings are either all positive or all negative (such a knot is called a special alternating knot), then its ropelength is bounded below by a constant multiple of its crossing number.
