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This article is cited in 1 scientific paper (total in 1 paper)
On the fundamental groups of complements of toral curves
Vik. S. Kulikov Moscow State University of Railway Communications
Abstract:
We show that for almost all curves $D$ in $\mathbb C^2$ given by an equation of the form
$g(x,y)^a+h(x,y)^b=0$, where $a>1$ and $b>1$ are coprime integers, the fundamental group of the complement of the curve has presentation $\pi_1(\mathbb C^2 \setminus D) \simeq (x_1,x_2\mid x_1^a=x_2^b)$, that is, it coincides with the group of the torus
knot $K_{a,b}$. In the projective case, for almost every curve $\overline D$
in $\mathbb P^2$ which is the projective closure of a curve in $\mathbb C^2$ given by an equation of the form $g(x,y)^a+h(x,y)^b=0$, the fundamental group $\pi_1(\mathbb P^2\setminus\overline D)$ of the complement is a free product with amalgamated subgroup of two cyclic groups of finite order. In particular, for the general curve
$\overline D\subset\mathbb P^2$ given by the equation
$l_{bc}^a(z_0,z_1,z_2)+l_{ac}^b(z_0,z_1,z_2)=0$, where $l_q$ is a homogenous polynomial of degree $q$, we have $\pi_1(\mathbb P^2\setminus\overline D)\simeq\langle x_1,x_2\mid x_1^a=x_2^b,x_1^{ac}=1\rangle$.
Received: 11.05.1995
Citation:
Vik. S. Kulikov, “On the fundamental groups of complements of toral curves”, Izv. Math., 61:1 (1997), 89–112
Linking options:
https://www.mathnet.ru/eng/im106https://doi.org/10.1070/IM1997v061n01ABEH000106 https://www.mathnet.ru/eng/im/v61/i1/p89
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Abstract page: | 610 | Russian version PDF: | 169 | English version PDF: | 18 | References: | 66 | First page: | 1 |
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