On the theory of Lucas coloring

TL;DR

This paper introduces Lucas-Coloring, a new graph coloring concept linked to planar graphs, which connects to link invariants, combinatorial objects like alternating sign matrices, and tiling enumeration formulas.

Contribution

It defines Lucas-Coloring for planar graphs, relates it to link invariants, and derives new combinatorial formulas for tilings and matchings.

Findings

Lucas-Coloring provides a numerical invariant of link diagrams.

Retrieves Alternating Sign Matrices as a special case of Lucas-Coloring.

Derives a summation formula for lozenge tilings related to Aztec Diamond enumeration.

Abstract

In this paper, 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, $Cob^{3}_{/l}$ . The Karoubi envelope of $Cob^{3}_{/l}$ 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, which concerns domino tilings of Aztec Diamonds.