In this paper we use a knot invariant, namely the Tristram--Levine signature, to study deformations of singular points of plane curves. We bound, in some cases, the difference between the $M$-number of the singularity of the central fiber and the sum of $M$-numbers of the generic fiber.
References
M. Borodzik: Morse theory for plane algebraic curves, J. Topol. 5 (2012), 341–365. MR2928080 10.1112/jtopol/jts006 M. Borodzik: Morse theory for plane algebraic curves, J. Topol. 5 (2012), 341–365. MR2928080 10.1112/jtopol/jts006
M. Borodzik: Abelian $\rho$-invariants of iterated torus knots; in Low-Dimensional and Symplectic Topology, Proc. Sympos. Pure Math. 82, Amer. Math. Soc., Providence, RI, 2011, 29–38. MR2768651 10.1090/pspum/082/2768651 M. Borodzik: Abelian $\rho$-invariants of iterated torus knots; in Low-Dimensional and Symplectic Topology, Proc. Sympos. Pure Math. 82, Amer. Math. Soc., Providence, RI, 2011, 29–38. MR2768651 10.1090/pspum/082/2768651
M. Borodzik: A $\rho$-invariant of iterated torus knots, arxiv:0906.3660v3, an updated version of [3251-03?] with two sign mistakes corrected. 0906.3660v3 0906.3660v3 M. Borodzik: A $\rho$-invariant of iterated torus knots, arxiv:0906.3660v3, an updated version of [3251-03?] with two sign mistakes corrected. 0906.3660v3 0906.3660v3
M. Borodzik and H. Żo\lądek: Complex Algebraic Plane Curves via Poincaré–Hopf Formula, III, Codimension Bounds, J. Math. Kyoto Univ. 48 (2008), 529–570. MR2511050 euclid.kjm/1250271383
M. Borodzik and H. Żo\lądek: Complex Algebraic Plane Curves via Poincaré–Hopf Formula, III, Codimension Bounds, J. Math. Kyoto Univ. 48 (2008), 529–570. MR2511050 euclid.kjm/1250271383
C. Christopher and S. Lynch: Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces, Nonlinearity 12 (1999), 1099–1112. MR1709857 10.1088/0951-7715/12/4/321 C. Christopher and S. Lynch: Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces, Nonlinearity 12 (1999), 1099–1112. MR1709857 10.1088/0951-7715/12/4/321
D. Eisenbud and W. Neumann: Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Annals of Mathematics Studies 110, Princeton Univ. Press, Princeton, NJ, 1985. MR817982 D. Eisenbud and W. Neumann: Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Annals of Mathematics Studies 110, Princeton Univ. Press, Princeton, NJ, 1985. MR817982
\begingroup G.-M. Greuel, C. Lossen and E. Shustin: Introduction to Singularities and Deformations, Springer Monographs in Mathematics, Springer, Berlin, 2007. \endgroup MR2290112 \begingroup G.-M. Greuel, C. Lossen and E. Shustin: Introduction to Singularities and Deformations, Springer Monographs in Mathematics, Springer, Berlin, 2007. \endgroup MR2290112
R. Kirby and P. Melvin: Dedekind sums, $\mu$-invariants and the signature cocycle, Math. Ann. 299 (1994), 231–267. MR1275766 10.1007/BF01459782 R. Kirby and P. Melvin: Dedekind sums, $\mu$-invariants and the signature cocycle, Math. Ann. 299 (1994), 231–267. MR1275766 10.1007/BF01459782
S.Yu. Orevkov: On rational cuspidal curves. I. Sharp estimate for degree via multiplicities, Math. Ann. 324 (2002), 657–673. MR1942244 10.1007/s002080000191 S.Yu. Orevkov: On rational cuspidal curves. I. Sharp estimate for degree via multiplicities, Math. Ann. 324 (2002), 657–673. MR1942244 10.1007/s002080000191
S.Yu. Orevkov, M.G. Zaidenberg: On the number of singular points of plane curves.. alg-geom/9507005 S.Yu. Orevkov, M.G. Zaidenberg: On the number of singular points of plane curves.. alg-geom/9507005
