On the theory of Lucas coloring

TL;DR

The paper introduces Lucas-Coloring, a two-color edge-coloring of planar graphs governed by Fibonacci-like local constraints, and proves a duality and a key statistic m(G) that aggregate colorings by the number of 'special' vertices. It then situates Lucas-Coloring inside Bar-Natan’s Cob^{3}_{/l} category via the Karoubi envelope, showing a one-to-one correspondence between Lucas-Colorings of a planar graph g and the direct summands in the Karoubi-augmented Khovanov-Lee complex, thereby linking graph colorings to topological link invariants. A central result is a bijection between n×n alternating sign matrices and restricted Lucas-Colorings of the grid graph G_n, providing a new combinatorial lens on ASM. The work further connects Lucas-Coloring to lozenge tilings and perfect matchings: it expresses the number of lozenge tilings of certain regions as a weighted sum over Lucas-Colorings, and builds a canonical graph whose perfect matchings enumerate Lucas-Colorings via m(g). An algebraic, state-sum proof using a Matching Algebra complements the combinatorial construction, illustrating a robust framework for translating global matching problems into local colorings. Overall, the paper bridges knot homology, ASM theory, and tiling combinatorics through the unifying notion of Lucas-Coloring."

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.

Figures (26)

• Figure 1: The relations in Bar-Natan's category $Cob^{3}_{/l}$
• Figure 2: The relations in Karoubi envelope $Kar(Cob^{3}_{/l})$
• Figure 3: The Aztec diamond of order $n$ for $n=1, 2, 3, 4$ and $5$.
• Figure 4: For $\triangle$ graph there are only two $Lucas-Colorings$.
• Figure 5: For Wheel graph $W_{5}$ graph there are $11$$Lucas-Colorings$.

Theorems & Definitions (29)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Remark 6
• Remark 7
• Lemma 1
• Lemma 2
• Lemma 3