Parity-of-stones determines the sign of the boundary monodromy

Determine whether, for every simply connected region R in the hexagonal grid that admits a signed tiling by the stone, bone, and snake tiles (with tiles of either sign added along the boundary), the SL2(C) boundary monodromy matrix ∂R equals (−1)^s · I, where s is the minimal number of stone tiles required in any such signed tiling and I is the 2×2 identity matrix.

Background

The paper introduces a necessary condition on signed tilings by assigning SL2(C) connection matrices (α, β, γ) to edges of the hexagonal grid and studying the matrix product ∂R around the boundary. It is shown that removing a bone or a snake tile does not change the product ∂R, while removing a stone multiplies ∂R by −1.

From this, Theorem 2.1 (and its corollaries) implies that any region R having a signed tiling by stones, bones, and snakes must have ∂R = ±I, and when a signed tiling uses only bones and snakes one necessarily has ∂R = I. The conjecture sharpens this by proposing that the sign of ∂R is exactly determined by the parity of the minimal number of stones required among all such signed tilings built along the boundary.