每个Plat表示都存在一个正的有界辫子代表元
Every Plat Presentation Admits a Positive Bounded Braid Representative
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中文总结 AI 辅助
该研究针对纽结分类中Hilden双陪集类庞大的问题，证明每个Hilden双陪集存在有界正辫子代表元，为桥有限性猜想研究提供了代数方法的建设性工具。
中文摘要 AI 辅助
通过n-桥位置对纽结类型进行分类，需要处理庞大的Hilden双陪集类H₂n\backslash B₂n/H₂n。我们通过证明每个Hilden双陪集都存在一个正辫子代表元，其Dehornoy最小值有明确界，从而对此加以限制。该证明依赖于关键观察Δ∈H₂n，这使我们能够利用Garside左贪婪正规形式从任意给定的辫子代表元中移除所有Δ的幂次。我们根据初始表示的Garside长度，给出了其Dehornoy最小值的显式上界。这个有界代表元限制了Plat表示的代数空间，为研究桥同痕类提供了潜在的计算工具，并为桥有限性猜想的代数方法提供了建设性步骤。
英文摘要
Classifying link types via their n-bridge positions requires navigating vast Hilden double coset classes H2n\B2n/H2n. We provide a restriction on this by proving that every Hilden double coset admits a positive braid representative whose Dehornoy minimum is explicitly bounded. The proof relies on the key observation that $Δ\in H_{2n}$, which allows us to utilize the Garside left-greedy normal form to remove all powers of $Δ$ from any given braid representative. We provide an explicit upper bound on its Dehornoy minimum in terms of the Garside length of the initial presentation. This bounded representative restricts the algebraic space of plat presentations, providing a potential computational tool for studying bridge isotopy classes and offering a constructive step toward algebraic approaches to the Bridge Finiteness Conjecture.
