#### Volume 3, issue 1 (2003)

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The universal order one invariant of framed knots in most $S^1$–bundles over orientable surfaces

Algebraic & Geometric Topology 3 (2003) 89–101
 arXiv: math.GT/0209027
##### Abstract

It is well-known that self-linking is the only $ℤ$–valued Vassiliev invariant of framed knots in ${S}^{3}$. However for most $3$–manifolds, in particular for the total spaces of ${S}^{1}$–bundles over an orientable surface $F\ne {S}^{2}$, the space of $ℤ$–valued order one invariants is infinite dimensional. We give an explicit formula for the order one invariant $I$ of framed knots in orientable total spaces of ${S}^{1}$–bundles over an orientable not necessarily compact surface $F\ne {S}^{2}$. We show that if $F\ne {S}^{2},{S}^{1}×{S}^{1},$ then $I$ is the universal order one invariant, i.e. it distinguishes every two framed knots that can be distinguished by order one invariants with values in an Abelian group.

##### Keywords
Goussarov–Vassiliev invariants, wave fronts, Arnold's invariants of fronts, curves on surfaces
Primary: 57M27
Secondary: 53D99