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1 Bulletin of the Polish Academy of Sciences. Mathematics

1 2020

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Symplectic structure on colorings, Lagrangian tangles and Tits buildings

Dymara Jan, Januszkiewicz Tadeusz, Przytycki Józef H.

Bulletin of the Polish Academy of Sciences. Mathematics

2020

Vol. 68, no. 2

169--194

We define a symplectic form ϕ on a free R-module R2n-2 associated to 2n points on a circle. Using this form, we establish a relation between submodules of R2n-2 induced by Fox R-colorings of an n-tangle and Lagrangians or virtual Lagrangians in the symplectic structure (R2n-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, every 2π/n-rotation of an n-tangle yields an isometry of the symplectic space R2n-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 S3 along a given link.