The universal order one invariant of framed knots in most \(S^1\)-bundles over orientable surfaces. (English) Zbl 1030.57019

Framed knots in an oriented circle bundle \(M\) over an oriented surface \(F\) are considered. For \(F\neq S^2\), a framed knot invariant \(I\) which takes values in the \({\mathbb Z}\)-module generated by the quotient set of \(\pi_1(M)\oplus \pi_1(M)\) modulo the actions of \(\pi_1(M)\) by conjugation and of \({\mathbb Z}_2\) by permutation of the summands is defined. This invariant is shown to have order one. For \(F\neq S^1\times S^1\), \(I\) is proved to be universal among first order invariants in the sense that if \(A\) is a first order invariant and \(A(K)\neq A(L)\) for some framed knots \(K\) and \(L\) then \(I(K)\neq I(L)\).


57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
53D99 Symplectic geometry, contact geometry
58K15 Topological properties of mappings on manifolds
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