## Some knot theory of complex plane curves (1983)

Venue: | Enseignement Mathématique |

Citations: | 12 - 2 self |

### BibTeX

@ARTICLE{Rudolph83someknot,

author = {Lee Rudolph},

title = {Some knot theory of complex plane curves},

journal = {Enseignement Mathématique},

year = {1983},

pages = {0106058}

}

### Years of Citing Articles

### OpenURL

### Abstract

for complex plane curves How can a complex curve be placed in a complex surface? The question is vague; many different ways to make it more specific may be imagined. The theory of deformations of complex structure, and their associated moduli spaces, is one way. Differential geometry and function theory, curvatures and currents, could be brought in. Even the generalized Nevanlinna theory of value distribution, for analytic curves, can somehow be construed as an aspect of the “placement problem”. By “knot theory ” I mean to connote those aspects of the situation that are more immediately topological. I hope to show that there is something of interest there. §2. A triptych. Here are three ways to interpret the phrase “knot theory of complex plane curves”. Globally: the “complex plane ” is projective space CP 2 or affine space C 2; a “curve ” is an algebraic curve (in projective space) or an algebraic or analytic curve (in affine space); here, “knot theory ” has historically been largely concerned with studying the “knot group”, though there are also results on “knot type”. Locally: a “complex plane curve ” is the germ of a plane curve (algebraic, analytic, or formal) over C; this is the study of singularities, and “knot theory ” has been the classical knot theory of links in the 3-sphere, put to work in the service of that study. In between: a “complex plane curve ” is an analytic curve in a reasonable open set in a complex surface (chiefly, in the theory as so far developed, the interior of a ball or a bidisk), well-behaved at the boundary; a knot-theorist can study either of two codimension-2 situations—the complex curve in its ambient space, or the boundary of this pair. This middle panel of the triptych has been less studied than the other two, though it is of obvious relevance to both.

### Citations

126 | Three-Dimensional Link Theory and Invariants of Plane Curve - Eisenbud, Newmann - 1985 |

67 | Gauge Theory for Embedded Surfaces
- Kronheimer, Mrowka
- 1993
(Show Context)
Citation Context ...ociated to any stratum incident to the given stratum. Partly, it was the desire to apply this fact to the proof of the Zariski Conjecture 2 Kronheimer and Mrowka, by proving the local Thom Conjecture =-=[K-M]-=-, answered Milnor’s question affirmatively. See also [Ru 7] for further knot-theoretical consequences of the truth of the local Thom Conjecture.6 LEE RUDOLPH (see below) which led investigators for m... |

61 |
Recognition of the problem
- Harris
- 1987
(Show Context)
Citation Context ...the knot group of an arbitrary curve [vK]; van Kampen gave his solution in terms of a certain presentation of the knot group. If Γ has (geometric) degree d, then van 3 As of 1986, a theorem of Harris =-=[Ha]-=-. 4 An alternative proof of a stronger theorem (the link at infinity may be any closed positive braid) was given in 1988 by Orevkov [O], using braid-theoretic methods related to those in [Ru 1].SOME ... |

47 |
Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace C 2
- Suzuki
- 1974
(Show Context)
Citation Context ...ion without local maxima on IntS, where N(z, w) = |z| 2 + |w| 2 ; and a surface-with-boundary S ∈ D4 r , with BdS = S3 r ∩S, is a ribbon surface if the inclusion (S, Bd S) ⊂ (D4 r , S3 r ) 5 See also =-=[Suz]-=-, for an analytic proof of the main theorem of [A-M], published slightly earlier. 6 At least two other topological proofs have since been given. That in [N-R] dispenses with knot cobordism and uses in... |

37 | Quasipositivity as an obstruction to sliceness - Rudolph - 1993 |

17 |
Complex algebraic plane curves via their links at infinity
- Neumann
- 1989
(Show Context)
Citation Context ... published slightly earlier. 6 At least two other topological proofs have since been given. That in [N-R] dispenses with knot cobordism and uses instead a notion of “unfolding a fibred knot”; that in =-=[N]-=- uses the calculus of splice diagrams. Both these proofs have the further virtue that they recover, not just the Abhyankar–Moh—Suzuki classification of polynomial embeddings of C in C 2 , but also the... |

17 |
An irreducible, simply connected algebraic curve in C 2 is equivalent to a quasihomogeneous curve
- Zaidenberg
- 1983
(Show Context)
Citation Context ...e that they recover, not just the Abhyankar–Moh—Suzuki classification of polynomial embeddings of C in C 2 , but also the Zaĭdenberg–Lin classification of singular polynomial injections of C in C 2 , =-=[Z-L]-=-.SOME KNOT THEORY OF COMPLEX PLANE CURVES 9 is isotopic through embeddings of pairs to a ribbon embedding. To demand that a surface be ribbon is to place genuine topological restrictions on the embed... |

14 |
Unfoldings in knot theory
- Neumann, Rudolph
- 1987
(Show Context)
Citation Context ...ion (S, Bd S) ⊂ (D4 r , S3 r ) 5 See also [Suz], for an analytic proof of the main theorem of [A-M], published slightly earlier. 6 At least two other topological proofs have since been given. That in =-=[N-R]-=- dispenses with knot cobordism and uses instead a notion of “unfolding a fibred knot”; that in [N] uses the calculus of splice diagrams. Both these proofs have the further virtue that they recover, no... |

11 |
Quasipositivité d’une courbe analytique dans une boule pseudo-convexe
- Boileau, Orevkov
(Show Context)
Citation Context ...sed curve, R the compact simply-connected region it bounds, F : R → En a continuous n-valued function analytic on IntR with 7 Our ignorance is now much less extensive. See footnote 2 and, especially, =-=[B-O]-=-.10 LEE RUDOLPH F(γ) ∩ ∆ = ∅. Then there is some radius M > 0 so that the graph of F |γ lies in the open solid torus γ × {w ∈ C : |w| < M}; and this graph is a (not necessarily connected) n-sheeted c... |

5 |
The commutant of the fundamental group of the complement of a plane algebraic curve, Russian Math. surveys 45
- Orevkov
- 1990
(Show Context)
Citation Context ...geometric) degree d, then van 3 As of 1986, a theorem of Harris [Ha]. 4 An alternative proof of a stronger theorem (the link at infinity may be any closed positive braid) was given in 1988 by Orevkov =-=[O]-=-, using braid-theoretic methods related to those in [Ru 1].SOME KNOT THEORY OF COMPLEX PLANE CURVES 7 Kampen’s presentation has d generators x1, . . . , xd which represent loops in a fixed projective... |