Symplectic structure on colorings, Lagrangian tangles and Tits buildings
Bulletin of the Polish Academy of Sciences Mathematics
We define a symplectic form ϕ on a free R-module R 2n−2 associated to 2n points on a circle. Using this form, we establish a relation between submodules of R 2n−2 induced by Fox R-colorings of an n-tangle and Lagrangians or virtual Lagrangians in the symplectic structure (R 2n−2 , ϕ) depending on whether R is a field or a PID. We prove that when R = Zp, p > 2, all Lagrangians are induced by Fox R-colorings of some n-tangles and note that for p = 2 and n > 3 this is no longer true. For any ring,
... true. For any ring, every 2π/nrotation of an n-tangle yields an isometry of the symplectic space R 2n−2 . We analyze invariant Lagrangian subspaces of this rotation and we partially answer the question whether an operation of rotation (generalized mutation) defined in [A-P-R] preserves the first homology group of the double branched cover of S 3 along a given link. 2020 Mathematics Subject Classification: Primary 57K10; Secondary 57R17, 51E24, 20E42. . ( 1 ) This relation might be viewed as a parallel of the well-known result that 3manifolds yield Lagrangians in H1(∂M, Q). ( 4 ) If R = Zp we write Colp(T ) instead Col Zp (T ).