Algebraic and Geometric Topology 3 (2003), paper no. 3, pages 89-101.

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

Vladimir Chernov (Tchernov)

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 not 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 not S^2. We show that if F is not S^2 or S^1 X 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

AMS subject classification. Primary: 57M27. Secondary: 53D99.

DOI: 10.2140/agt.2003.3.89

E-print: arXiv:math.GT/0209027

Submitted: 3 December 2002. Accepted: 23 January 2003. Published: 28 January 2002.

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Vladimir Chernov (Tchernov)
Department of Mathematics, 6188 Bradley Hall
Dartmouth College, Hanover, NH 03755, USA

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