(1, 1)-knots via the mapping class group of the twice punctured torus

doi:10.1515/advg.2004.016

Cited by 25 publications

(25 citation statements)

References 22 publications

(35 reference statements)

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“…where Σ 1 is the torus S 1 × S 1 ( [Schn03], [CattMu04]). This implies that we can define actions of these groups on the set of generating vectors for G of type (0 | m 1 , .…”

Section: Moduli Spacesmentioning

confidence: 99%

The classification of surfaces with p_g = q = 1 isogenous to a product of curves

Carnovale

Polizzi

2009

Advg

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Abstract. A smooth, projective surface S is said to be isogenous to a product if there exist two smooth curves C, F and a finite group G acting freely on C × F so that S = (C × F )/G. In this paper we classify all surfaces with pg = q = 1 which are isogenous to a product. IntroductionThe classification of smooth, complex surfaces S of general type with small birational invariants is quite a natural problem in the framework of algebraic geometry. For instance, one may want to understand the case where the Euler characteristic χ(O S ) is 1, that is, when the geometric genus p g (S) is equal to the irregularity q(S). All surfaces of general type with these invariants satisfy p g ≤ 4. In addition, if p g = q = 4 then the self-intersection K 2 S of the canonical class of S is equal to 8 and S is the product of two genus 2 curves, whereas if p g = q = 3 then K 2 S = 6 or 8 and both cases are completely described. On the other hand, surfaces of general type with p g = q = 0, 1, 2 are still far from being classified. We refer the reader to the survey paper [BaCaPi06] for a recent account on this topic and a comprehensive list of references. A natural way of producing interesting examples of algebraic surfaces is to construct them as quotients of known ones by the action of a finite group. For instance Godeaux constructed in [Go31] the first example of surface of general type with vanishing geometric genus taking the quotient of a general quintic surface of P 3 by a free action of Z 5 . In line with this Beauville proposed in [Be96, p. 118] the construction of a surface of general type with p g = q = 0, K 2 S = 8 as the quotient of a product of two curves C and F by the free action of a finite group G whose order is related to the genera g(C) and g(F ) by the equality |G| = (g(C) − 1)(g(F ) − 1). Generalizing Beauville's example we say that a surface S is isogenous to a product if S = (C × F )/G, for C and F smooth curves and G a finite group acting freely on C × F . A systematic study of these surfaces has been carried out in [Ca00]. They are of general type if and only if both g(C) and g(F ) are greater than or equal to 2 and in this case S admits a unique minimal realization where they are as small as possible. From now on, we tacitly assume that such a realization is chosen, so that the genera of the curves and the group G are invariants of S. The action of G can be seen to respect the product structure on C × F . This means that such actions fall in two cases: the mixed one, where there exists some element in G exchanging the two factors (in this situation C and F must be isomorphic) and the unmixed one, where G acts faithfully on both C and F and diagonally on their product. After [Be96], examples of surfaces isogenous to a product with p g = q = 0 appeared in [Par03] and [BaCa03], and their complete classification was obtained in [BaCaGr06]. The next natural step is therefore the analysis of the case p g = q = 1. Surfaces of general type with these invariants are the irregular ones with the lowest geometric genus ...

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“…where Σ 1 is the torus S 1 × S 1 ( [Schn03], [CattMu04]). This implies that we can define actions of these groups on the set of generating vectors for G of type (0 | m 1 , .…”

Section: Moduli Spacesmentioning

confidence: 99%

The classification of surfaces with p_g = q = 1 isogenous to a product of curves

Carnovale

Polizzi

2009

Advg

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“…Let K be a (g, 1)-knot in a 3-manifold N with first homology group given by (3). Then K admits n-fold strongly-cyclic branched coverings if and only if gcd(e i , n) = gcd(e ′ i , n), for each i = 1, .…”

Section: Strongly-cyclic Branched Coverings Of Knotsmentioning

confidence: 99%

“…Any knot K in a 3-manifold N admits a (g, 1)-decomposition, for a certain g ≥ h(N), where h(N) is the Heegaard genus of N. In the following a knot admitting a (g, 1)-decomposition will be called a (g, 1)-knot. The particular case of (1, 1)-knots has recently been investigated in several papers from different points of view (see references in [3]). …”

Section: Introductionmentioning

confidence: 99%

Strongly-cyclic branched coverings of knots via (g, 1)-decompositions

2007

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“…Their results have been included in an organic and more general context in [11], where it is proved that the fundamental group of each n-fold strongly-cyclic branched covering of a (1, 1)-knot admits a cyclic presentation encoded by a genus n Heegaard diagram. In [12] this result has been improved, yielding a constructive algorithm that explicitly gives the cyclic presentation, starting from a representation of the (1, 1)-knot via the mapping class group of the twice punctured torus (see [13] for further details on this representation).…”

Section: Introductionmentioning

confidence: 99%