论文标题
非循环负圆环
Non-loose negative torus knots
论文作者
论文摘要
我们在$ 3 $ -sphere上所有接触结构中研究了负圆环结的Legendrian和横向实现。我们对强烈的非循环横向实现和强烈的非循环legendrian实现进行了完整的分类，而瑟斯顿·奔驰的不变型小于$ -pq $。 此外，我们表明，强烈的非循环横向实现$ t $由其非零的不变式$ \ mathfrak t（t）$在Shonus Floer同源性中。但是，并非可以实现$ hfk^ - （t _ {（p，q）}）$的所有元素。 在此过程中，我们将我们的传奇人物实现与沿着传奇手术的紧密接触结构联系起来。具体来说，我们意识到镜头空间上的所有紧密结构$ l（PQ+1，P^2）$作为单一的Legendrian手术，涉及Legendrian $ t _ {（P，-Q）} $，并且我们将跨越结构的横向实现与沿着小规模构成的大型负面源头上的不可填充的紧密结构相关联。
We study Legendrian and transverse realizations of the negative torus knots $T_{(p,-q)}$ in all contact structures on the $3$-sphere. We give a complete classification of the strongly non-loose transverse realizations and the strongly non-loose Legendrian realizations with the Thurston-Bennequin invariant smaller than $-pq$. Additionally, we show that the strongly non-loose transverse realizations $T$ are classified by their non-zero invariants $\mathfrak T(T)$ in the minus version of the knot Floer homology. However, not all the elements of $HFK^-(T_{(p,q)})$ can be realized. Along the way, we relate our Legendrian realizations to the tight contact structures on the Legendrian surgeries along them. Specifically, we realize all tight structures on the lens spaces $L(pq+1,p^2)$ as a single Legendrian surgery on a Legendrian $T_{(p,-q)}$, and we relate transverse realizations in overtwisted structures to the non-fillable tight structures on the large negative surgeries along the underlying knots.
