The Alexander polynomial of a plane curve singularity via the ring of functions on it

Campillo, A.; Delgado, F.; Guseĭn-Zade, S. M.

doi:10.1215/s0012-7094-03-11712-3

Cited by 74 publications

(92 citation statements)

References 12 publications

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“…If (S, 0) is smooth or if it is the E 8 -surface singularity, the W-Poincaré series of the collection {v i } coincides with the usual Poincaré series of {v i } described above: [2], [5], [9]. Remark 2.…”

Section: The Weil-poincaré Seriesmentioning

confidence: 91%

Weil-Poincaré series and topology of collections of valuations on rational double points

Campillo¹,

Delgado²,

Guseĭn-Zade³

2021

Preprint

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Earlier it was described to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. It was shown that, except some explicitly described cases, the Alexander polynomial of an algebraic link determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere coincides with the Poincaré series of the corresponding set of curve valuations. The latter one can be defined as an integral over the space of divisors on the E 8 -singularity.Here we consider a similar integral for rational double point surface singularities over the space of Weil divisors called the Weil-Poincaré series. We show that, except a few explicitly described cases the Weil-Poincaré series of a collection of curve valuations on a rational double point surface singularity determines the topology of the corresponding link. We give analogous statements for collections of divisorial valuations.

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Section: The Weil-poincaré Seriesmentioning

confidence: 91%

Weil-Poincaré series and topology of collections of valuations on rational double points

Campillo¹,

Delgado²,

Guseĭn-Zade³

2021

Preprint

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“…В качестве примера вычисления с помощью интегрирования по эйлеровой характеристике опишем вычисление ряда Пуанкаре фильтрации на кольце O C 2 ,0 ростков функций от двух переменных, определенной компонентами ростка плоской кривой: [28], [24].…”

Section: вычисление рядов пуанкаре через интегралы по эйлеровой харакunclassified

“…Теорема 6 [28], [24]. В терминах разрешения особенности кривой C ряд Пуанкаре фильтрации, соответствующей кривой, дается формулой…”

Section: вычисление рядов пуанкаре через интегралы по эйлеровой харакunclassified

“…Следствие [31], [32], [28]. Ряд Пуанкаре (одноиндексной) фильтрации, соответствующей неприводимой кривой C , совпадает с многочленом Александера алгебраического узла, соответствующего особенности кривой C , деленным на 1 − t. При r > 1 ряд Пуанкаре мультииндексной фильтрации, со-…”

Section: теорема 6 доказанаunclassified

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Integration with respect to the Euler characteristic and its applications

Guseĭn-Zade¹

2010

Uspekhi Mat. Nauk

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Обсуждается понятие интегрирования по эйлеровой характеристике и его обобщения: интегрирование по бесконечномерным пространствам дуг или функций, мотивное интегрирование. Описываются приложения этих понятий к вычислению дзета-функций монодромий, рядов Пуанкаре мультииндексных фильтраций, производящих рядов классов некоторых классифицирующих пространств и т. д. Библиография: 70 названий.

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“…But the subject has not stopped there. Very recent papers include A'Campo's beautiful construction of the Milnor fibration from a real morsification [1], and an intriguing and surprising geometric formula for the Alexander polynomial in [17].…”

Section: Question What Invariant or Collection Of Invariants Completmentioning

confidence: 99%

Topology of Hypersurface Singularities

Neumann¹

2003

Mathematische Werke / Mathematical Works

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Kähler's paperÜber die Verzweigung einer algebraischen Funktion zweier Veränderlichen in der Umgebung einer singulären Stelle" offered a more perceptual view of the link of a complex plane curve singularity than that provided shortly before by Brauner. Kähler's innovation of using a "square sphere" became standard in the toolkit of later researchers on singularities. We describe his contribution and survey developments since then, including a brief discussion of the topology of isolated hypersurface singularities in higher dimension.Key words and phrases. plane curve singularity, link of a singularity, splice diagram.

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The Alexander polynomial of a plane curve singularity via the ring of functions on it

Cited by 74 publications

References 12 publications

Weil-Poincaré series and topology of collections of valuations on rational double points

Weil-Poincaré series and topology of collections of valuations on rational double points

Integration with respect to the Euler characteristic and its applications

Topology of Hypersurface Singularities

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