There are oriented 2-knots with isomorphic fundamental groups, semilinearly isomorphic second homotopy modules, and isomorphic quandles, but inequivalent first Postnikov invariants.
The quandle and group for higher-dimensional and virtual knots
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abstract
Joyce has shown that the fundamental quandle of a classical knot can be derived from consideration of the fundamental group and the peripheral structure of the knot, and also that the group and much of the peripheral structure can be recovered from the quandle. We generalize these results to arbitrary dimensions, and also to virtual and welded knots and arcs.
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2026 1verdicts
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Fundamental Quandles Do Not Determine the First Postnikov Invariant of 2-Knots
There are oriented 2-knots with isomorphic fundamental groups, semilinearly isomorphic second homotopy modules, and isomorphic quandles, but inequivalent first Postnikov invariants.