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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
The universal order one invariant of framed knots in most $S^1$–bundles over orientable surfaces

Vladimir V Chernov

Algebraic & Geometric Topology 3 (2003) 89–101

arXiv: math.GT/0209027

Abstract

It is well-known that self-linking is the only –valued Vassiliev invariant of framed knots in S3. However for most 3–manifolds, in particular for the total spaces of S1–bundles over an orientable surface FS2, the space of –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 S1–bundles over an orientable not necessarily compact surface FS2. We show that if FS2,S1 × S1, 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
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 53D99
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Publication
Received: 3 December 2002
Accepted: 23 January 2003
Published: 28 January 2003
Authors
Vladimir V Chernov
Department of Mathematics
6188 Bradley Hall
Dartmouth College
Hanover NH 03755
USA