Gordian 分裂链在 Gehring 绳长问题中的研究
Gordian split links in the Gehring ropelength problem
AI总结：
该研究在允许分支间长度交换的宽松情形下，证明存在Gordian分裂链，还构造了无法通过粗同痕分裂的2分支Gordian平凡链，首次证实Gehring绳长存在局部极小值。
摘要：
粗链是指三维欧几里得空间中，链的每个分支与其他所有分支的距离至少为1的链。作为粗度概念的强化，厚嵌入链是一种粗链，其所有分支的半径为1/2的开法向圆盘丛均为嵌入。粗同痕是指粗链的链同痕，在整个过程中保持粗度和总长度不变；厚同痕是指厚嵌入链的链同痕，在整个过程中保持厚嵌入性和总长度不变。我们构造了一个同痕意义下的 Gordian 分裂链，即具有4个分支的厚嵌入链，它在拓扑上是可分裂的，但无法通过厚同痕实现分裂。这是首次在允许分支间长度交换的最宽松情形下证明存在 Gordian 分裂链。随后我们首次证明 Gehring 绳长存在局部而非全局极小值，具体而言，我们构造了一个2分支的同伦意义下的 Gordian 平凡链，即属于平凡链的链同伦类，但无法通过任何粗同痕实现分裂。
英文摘要：
A thick link is a link in $\mathbb{R}^3$ such that each component of the link lies at distance at least $1$ from every other component. Strengthening the notion of thickness, a thickly embedded link is a thick link whose open radius-$\tfrac{1}{2}$ normal disk bundles of all components are embedded. A thick homotopy is a link homotopy of a thick link that preserves thickness and total length throughout. A thick isotopy is a link isotopy of a thickly embedded link that preserves thick-embeddedness and total length throughout. We construct an isotopically Gordian split link, that is, a thickly embedded 4-component link which is topologically split but which cannot be split by a thick isotopy. This is the first time a Gordian split link is shown to exist in this most permissive setting where length trading between components is allowed. We then prove for the first time that local, non-global minima for Gehring ropelength exist. In particular, we construct a 2-component homotopically Gordian unlink, that is, a link in the link homotopy class of the unlink which cannot be split by any thick homotopy.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
