On the Legendrian Invariant in Knot Lattice Homology

The Ozsváth–Szabó contact invariant c + (𝜉) ∈ 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 HF + (− Y ) can be equipped with an Alexander grading, and we can define an element L ( L ) in the bigraded theory 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.