The universal order one invariant of framed knots in most S1–bundles over orientable surfaces

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 $F \neq S^{2}$ , 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 $F \neq S^{2}$ . We show that if $F \neq S^{2}, S^{1} \times S^{1},$ 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

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 53D99

References

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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

Volume 3, issue 1 (2003)

Vladimir V Chernov