The fundamental group of the complement of an algebraic curve

1974 ◽  
Vol 14 (2) ◽  
pp. 163-172 ◽  
Author(s):  
David Prill
Keyword(s):  
Fundamental Group ◽  
Algebraic Curve

On the Fundamental Group of an Algebraic Curve

1933 ◽  
Vol 55 (1/4) ◽  
pp. 255 ◽  
Author(s):  
Egbert R. Van Kampen
Keyword(s):  
Fundamental Group ◽  
Algebraic Curve

On the fundamental group of the complement of certain singular plane curves

1987 ◽  
Vol 102 (3) ◽  
pp. 453-457 ◽  
Author(s):  
András Némethi
Keyword(s):  
Projective Space ◽  
Fundamental Group ◽  
Algebraic Curve ◽  
Plane Curves ◽  
Image Position

Let C be a complex algebraic curve in the projective space ℙ2. The purpose of this paper is to calculate the fundamental group G of the complement of C in the case when C = X ∩ H1 ∩ … ∩ Hn−2, whereand Hi are generic hyperplanes (i = 1, … n − 2).

Invariant groups associated with an algebraic surface

1940 ◽  
Vol 36 (4) ◽  
pp. 414-423 ◽  
Author(s):  
D. B. Scott
Keyword(s):  
Fundamental Group ◽  
Algebraic Curve ◽  
Algebraic Surface ◽  
Algebraic Surfaces ◽  
Topological Invariants ◽  
Period Matrix ◽  
Birational Transformations

Alexander (1, 2) has introduced certain topological invariants of a manifold which arise from the intersections of cycles of non-complementary dimensions, and he points out that they are not derivable from the Betti and torsion numbers, nor from the fundamental group. In the present paper we consider some topological invariants of this type on an algebraic surface, and, although we cannot define them completely, we show that they are intimately connected with the multiplications of the period matrix of the simple integrals of the first kind. We are then led to a concept which we call the “intersection group” of the surface, which is, by its definition, topologically invariant, and we show that it is also invariant under birational transformations. The proofs are based on Lefschetz's theory of cycles for an algebraic surface (4) and some simple properties of the period matrix of an algebraic curve. The results obtained here have a number of applications to the theory of ∞3 correspondences between algebraic surfaces, as we propose to show in a later paper.

ON THE ARITHMETIC 2-BRIDGE KNOTS AND LINK ORBIFOLDS AND A NEW KNOT INVARIANT

1995 ◽  
Vol 04 (01) ◽  
pp. 81-114 ◽  
Author(s):  
HUGH M. HILDEN ◽  
MARIA TERESA LOZANO ◽  
JOSE MARIA MONTESINOS-AMILIBIA
Keyword(s):  
Fundamental Group ◽  
Algebraic Number ◽  
Algebraic Curve ◽  
Minimal Polynomial ◽  
Isotropy Group ◽  
Singular Set ◽  
Polynomial Invariant ◽  
Open Problems ◽  
Knot Invariant ◽  
Hyperbolic Orbifold

Let (p/q, n) denote the orbifold with singular set the two bridge knot or link p/q and isotropy group cyclic of orden n. An algebraic curve [Formula: see text] (set of zeroes of a polynomial r(x, z)) is associated to p/q parametrizing the representations of [Formula: see text] in PSL [Formula: see text]. The coordinates x, z, are trace(A2)=x, trace(AB)=z where A and B[Formula: see text] are the images of canonical generators a, b of [Formula: see text]. Let (xn, zn) be the point of [Formula: see text] corresponding to the hyperbolic orbifold (p/q, n). We prove the following result: The (orbifold) fundamental group of (p/q, n) is arithmetic if and only if the field Q(xn, zn) has exactly one complex place and ϕ(xn)<ϕ(zn)<2 for every real embedding [Formula: see text]. Consider the angle α for which the cone-manifold (p/q, α) is euclidean. We prove that 2cosα is an algebraic number. Its minimal polynomial (called the h-polynomial) is then a knot invariant. We indicate how to generalize this h-polynomial invariant for any hyperbolic knot. Finally, we compute h-polynomials and arithmeticity of (p/q, n) with p≦40, and (p/q, n) with p≦99q2≡1 mod p. We finish the paper with some open problems.

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