Symplectic structure on colorings, Lagrangian tangles and Tits buildings

We define a symplectic form $\widehat \varphi $ 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},\widehat \varphi ) $ depending on whether $R$ is a field or a PID. We prove that when $R=\mathbb {Z}_{p}$, $p \gt 2$, all Lagrangians are induced by Fox $R$-colorings of some $n$-tangles and note that for $p=2$ and $n \gt 3$ this is no longer true. For any ring, every $2\pi /n$-rotation 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 by Anstee et al. (1989) preserves the first homology group of the double branched cover of $S^{3}$ along a given link.