2026/07/23 by Sarah Zampa
#math.GT
The Ozsváth-Szabó contact invariant c+(ξ)\inHF+(-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.