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Symplectic structure on colorings, Lagrangian tangles and Tits buildings

2021
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Bulletin of the Polish Academy of Sciences Mathematics
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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,

doi:10.4064/ba201230-2-3
fatcat:zwctwyf7uffxpnrtre23iquuiq