On the Legendrian invariant in knot lattice homology
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.