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 $$I(K)\neq I(L)$$.

MSC:

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

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