On the Theory of Lucas Coloring

In this talk, we introduce the notion of "Lucas‑Coloring" associated with a planar graph g. When g is a 4‑regular, the enumeration of Lucas‑Coloring has an interesting interpretation. Specifically, it yields a numerical invariant of the associated Khovanov‑Lee complex of any link diagram D whose projection is equal to g. This complex resides in the Karoubi envelope of Bar‑Natan's formal cobordism category, Cob³/l. The Karoubi envelope of Cob³/l was introduced by Bar‑Natan and Morrison to provide a conceptual proof of Lee’s theorem. As an application of "Lucas‑Coloring", we first show how the Alternating Sign Matrices can be retrieved as a special case of Lucas‑Coloring. Next, we show a certain statistic on the Lucas‑Coloring enumerates the perfect matchings of a canonically defined graph on g. This construction allowed us to derive a summation formula of the enumeration of lozenge tilings of the region constructed out of a regular hexagon by removing the "maximal staircase" from its alternating corners in terms of powers of 2. This formula is reminiscent of the celebrated Aztec Diamond Theorem of Elkies, Kuperberg, Larsen, and Propp. 

This talk was originally presented at 42 YEARS OF ALTERNATING SIGN MATRICES Conference, In University of Ljubljana, Slovenia, on September 22, 2025. Funds for conference travel were provided by the Office of The Graduate School and Research, Ahmedabad University.