AI 中文总结
研究纽结格同调中的勒让德不变量，通过将正规曲面奇点链环的接触不变量转换到格同调理论，利用亚历山大分次定义元素\(\mathcal{L}(L)\)，证明其在爆破下不变，并借助链复形的滤过链同伦给出勒让德不变量的部分组合描述。
AI 中文摘要
配备其典范接触结构\((Y,\xi)\)的正规曲面奇点的链环的奥兹瓦思 - 萨博接触不变量\(c^+(\xi)\in\mathrm{HF}^+(-Y)\)，被博德纳 - 普拉梅涅夫斯卡娅转换到格同调理论。考虑链环中的横向代数纽结\(L\)时，计算\(\mathrm{HF}^+(-Y)\)的链复形可配备亚历山大分次，能在双分次理论\(\mathrm{HFK}^+(-Y,L)\)中定义元素\(\mathcal{L}(L)\)，它通过忘记滤过映射到接触元素。我们证明该元素的亚历山大分次（由奥兹瓦思 - 斯蒂普西茨 - 萨博定义）在基础配管图的所有爆破下不变。此外，利用特定类型爆破下所得格链复形是滤过链同伦的，链映射将一个链复形中的此元素映射到另一个，从而给出勒让德不变量的部分组合描述。
英文摘要
The Ozsváth-Szabó contact invariant $c^+(ξ)\in\mathrm{HF}^+(-Y)$ of the link of a normal surface singularity equipped with its canonical contact structure $(Y,ξ)$ was transposed to lattice homology theory by Bodnár-Plamenevskaya. When considering a transverse algebraic knot $L$ in the link, the chain complex computing $\mathrm{HF}^+(-Y)$ can be equipped with an Alexander grading, and we can define an element $\mathcal{L}(L)$ in the bigraded theory $\mathrm{HFK}^+(-Y,L)$, which maps to the contact element by forgetting the filtration. We show that the Alexander grading (as defined by Ozsváth-Stipsicz-Szabó) of this element is invariant under all blow-ups of the underlying plumbing graph. Furthermore, we utilize the fact that for specific types of blow-ups, the resulting lattice chain complexes are filtered chain homotopic and the chains maps map this element in one chain complex to the other, thereby providing a partial combinatorial description of the Legendrian invariant.
Comments10 pages, 1 figure, accepted for publication
