Geodesic
The universal order one invariant of framed knots in most S1–bundles over orientable surfaces
Chernov, Vladimir V1
1Department of Mathematics, 6188 Bradley Hall, Dartmouth College, Hanover NH 03755, USA
Algebraic and Geometric Topology, Tome 3 (2003) no. 1, pp. 89-101
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It is well-known that self-linking is the only ℤ–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 F≠S2, 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 S1–bundles over an orientable not necessarily compact surface F≠S2. We show that if F≠S2,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.

DOI : 10.2140/agt.2003.3.89
Keywords: Goussarov–Vassiliev invariants, wave fronts, Arnold's invariants of fronts, curves on surfaces

Chernov, Vladimir V  1

1 Department of Mathematics, 6188 Bradley Hall, Dartmouth College, Hanover NH 03755, USA 
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Chernov, Vladimir V. The universal order one invariant of framed knots in most S1–bundles over orientable surfaces. Algebraic and Geometric Topology, Tome 3 (2003) no. 1, pp. 89-101. doi: 10.2140/agt.2003.3.89

[1] L V Ahlfors, L Sario, Riemann surfaces, Princeton Mathematical Series 26, Princeton University Press (1960)

[2] V I Arnol’D, Invariants and perestroikas of fronts on a plane, Trudy Mat. Inst. Steklov. 209 (1995) 14

[3] M P Do Carmo, Riemannian geometry, Mathematics: Theory Applications, Birkhäuser (1992)

[4] V Tchernov, Shadows of wave fronts and Arnold–Bennequin type invariants of fronts on surfaces and orbifolds, from: "Differential and symplectic topology of knots and curves", Amer. Math. Soc. Transl. Ser. 2 190, Amer. Math. Soc. (1999) 153

[5] V Tchernov, Arnold-type invariants of wave fronts on surfaces, Topology 41 (2002) 1

[6] V Chernov, Y Rudyak, Affine linking numbers and causality relations for wave fronts,

[7] M Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 9, Springer (1986)

[8] W S Massey, Algebraic topology: an introduction, Graduate Texts in Mathematics 56, Springer (1977)

[9] M Polyak, On the Bennequin invariant of Legendrian curves and its quantization, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 77

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