#### 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.

Notes on file formats
Vladimir Chernov (Tchernov)

Department of Mathematics, 6188 Bradley Hall

Dartmouth College, Hanover, NH 03755, USA

Email: Vladimir.Chernov@dartmouth.edu

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