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| [[Image:Blue Figure-Eight Knot.png|56px|thumb|[[Figure-eight knot (mathematics)|Figure-eight knot]] ''is'' fibered.]]
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| In [[knot theory]], a branch of [[mathematics]], a [[knot (mathematics)|knot]] or [[link (knot theory)|link]] <math>K</math>
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| in the [[3-sphere|3-dimensional sphere]] <math>S^3</math> is called '''fibered''' or '''fibred''' if there is a 1-parameter family <math>F_t</math> of [[Seifert surface]]s for <math>K</math>, where the parameter <math>t</math> runs through the points of the [[unit circle]] <math>S^1</math>, such that if <math>s</math> is not equal to <math>t</math>
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| then the intersection of <math>F_s</math> and <math>F_t</math> is exactly <math>K</math>.
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| For example:
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| * The [[unknot]], [[trefoil knot]], and [[figure-eight knot (mathematics)|figure-eight knot]] are fibered knots.
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| * The [[Hopf link]] is a fibered link.
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| Fibered knots and links arise naturally, but not exclusively, in [[complex algebraic geometry]]. For instance, each [[Mathematical singularity|singular point]] of a [[complex plane curve]] can be described
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| topologically as the [[cone (topology)|cone]] on a fibered knot or link called the '''link of the singularity'''. The [[trefoil knot]] is the link of the [[Cusp (singularity)|cusp singularity]] <math>z^2+w^3</math>; the Hopf link (oriented correctly) is the link of the [[Singular point of a curve|node singularity]] <math>z^2+w^2</math>. In these cases, the family of Seifert surfaces is an aspect of the [[Milnor fibration]] of the singularity.
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| A knot is fibered if and only if it is the binding of some [[open book decomposition]] of <math>S^3</math>.
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| ==Knots that are not fibered==
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| [[Image:Knot-stevedore-sm.png|thumb|[[Stevedore knot (mathematics)|Stevedore's knot]] is ''not'' fibered]]
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| The [[Alexander polynomial]] of a fibered knot is monic, i.e. the coefficients of the highest and lowest powers of ''t'' are plus or minus 1. Examples of knots with nonmonic Alexander polynomials abound, for example the [[twist knot]]s have Alexander polynomials ''qt'' − (2''q'' + 1) + ''qt''<sup>−1</sup>, where ''q'' is the number of half-twists. [http://arxiv.org/abs/dg-ga/9612014] In particular the [[Stevedore knot (mathematics)|Stevedore's knot]] is not fibered.
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| ==See also==
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| *[[(−2,3,7) pretzel knot]]
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| ==References==
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| http://www.sciencedirect.com/science/article/pii/004093838290009X | |
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| http://www.msp.warwick.ac.uk/gt/2010/14-04/p050.xhtml | |
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| {{Knot theory}}
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| [[Category:Fibered knots and links| ]]
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| {{knottheory-stub}}
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