Casson invariant of knots associated with divides

Shumakovich, A.

doi:10.1090/S1061-0022-04-00826-X

Casson invariant of knots associated with divides
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by A. Shumakovich

St. Petersburg Math. J. 15 (2004), 625-637

DOI: https://doi.org/10.1090/S1061-0022-04-00826-X

Published electronically: July 7, 2004

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Abstract:

A formula for the Casson invariant of knots associated with divides is presented. The formula is written in terms of Arnold’s invariants of pieces of the divide. Various corollaries are discussed.

References

Norbert A’Campo, Le groupe de monodromie du déploiement des singularités isolées de courbes planes. II, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 395–404. MR 0435443
Norbert A’Campo, Real deformations and complex topology of plane curve singularities, Ann. Fac. Sci. Toulouse Math. (6) 8 (1999), no. 1, 5–23 (English, with English and French summaries). MR 1721511
Norbert A’Campo, Generic immersions of curves, knots, monodromy and Gordian number, Inst. Hautes Études Sci. Publ. Math. 88 (1998), 151–169 (1999). MR 1733329
Norbert A’Campo, Planar trees, slalom curves and hyperbolic knots, Inst. Hautes Études Sci. Publ. Math. 88 (1998), 171–180 (1999). MR 1733330
F. Aicardi, Tree-like curves, Singularities and bifurcations, Adv. Soviet Math., vol. 21, Amer. Math. Soc., Providence, RI, 1994, pp. 1–31. MR 1310594
V. I. Arnol′d, Plane curves, their invariants, perestroikas and classifications, Singularities and bifurcations, Adv. Soviet Math., vol. 21, Amer. Math. Soc., Providence, RI, 1994, pp. 33–91. With an appendix by F. Aicardi. MR 1310595
S. Chmutov, An integral generalization of the Gusein-Zade–Natanzon theorem, arXiv: math.GT/0207238.
S. Chmutov and S. Duzhin, Explicit formulas for Arnold’s generic curve invariants, The Arnold-Gelfand mathematical seminars, Birkhäuser Boston, Boston, MA, 1997, pp. 123–138. MR 1429890
William Gibson and Masaharu Ishikawa, Links of oriented divides and fibrations in link exteriors, Osaka J. Math. 39 (2002), no. 3, 681–703. MR 1932288
Mikami Hirasawa, Visualization of A’Campo’s fibered links and unknotting operation, Proceedings of the First Joint Japan-Mexico Meeting in Topology (Morelia, 1999), 2002, pp. 287–304. MR 1903697, DOI 10.1016/S0166-8641(01)00124-9
Michael Polyak, Invariants of curves and fronts via Gauss diagrams, Topology 37 (1998), no. 5, 989–1009. MR 1650406, DOI 10.1016/S0040-9383(97)00013-X
A. Shumakovich, Explicit formulas for the strangeness of plane curves, Algebra i Analiz 7 (1995), no. 3, 165–199 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 7 (1996), no. 3, 445-472. MR 1353494
O. Ya. Viro, First degree invariants of generic curves on surfaces, Preprint no. 21, Dep. Math. Uppsala Univ., Uppsala, 1994.