勒让德纽结的特征斜率刻画
Characterizing slopes for Legendrian knots
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中文总结 AI 辅助
该研究在勒让德纽结同痕类由经典不变量唯一确定的假设下，建立了光滑与接触特征斜率的关联准则，并将其应用于多种纽结的勒让德代表的特征斜率研究。
中文摘要 AI 辅助
我们在唯一性假设下建立了联系光滑特征斜率与接触特征斜率的准则。设$L$是$S^3$中标准接触结构下纽结$K$的勒让德代表，且假设$L$的同痕类由其经典不变量唯一确定：即瑟斯顿-本内奎不变量$tb(L)$与旋转数。对任意非零有理数$r$，若$r+tb(L)$是$K$的光滑特征斜率，则它成为$L$的接触特征斜率。作为应用，我们研究了未打结、三叶结、八字结、五叶结、$5_2$及$\boldsymbol{\bar{5_2}}$的勒让德代表的特征斜率。
英文摘要
We establish a criterion relating smooth and contact characterizing slopes under a uniqueness assumption. Let $L$ be a Legendrian representative of a knot $K\subset S^3$ with standard contact structure, and assume that the isotopy class of $L$ is uniquely determined by its classical invariants: the Thurston--Bennequin invariant $tb(L)$ and the rotation number. Then, for any non-zero rational number $r$, if $r+tb(L)$ is a smooth characterizing slope for $K$, it becomes a contact characterizing slope for $L$. As applications, we study the characterizing slopes for Legendrian representatives of the unknot, trefoil, figure-eight knot, cinquefoil, $5_2$, and $\overline{5_2}$.
