Editing Incompressible surface (section)

==Seifert surfaces==

A [[Seifert surface]] {{math|''S''}} for an oriented [[Link (knot theory)|link]] {{math|''L''}} is an [[oriented]] surface whose boundary is {{math|''L''}} with the same induced orientation.  If {{math|''S''}} is not {{math|''&pi;''<sub>1</sub>}}-injective in {{math|''S''<sup>3</sup> &minus; ''N''(''L'')}}, where {{math|''N''(''L'')}} is a [[tubular neighborhood]] of ''L'', then the loop theorem gives a compressing disk that one may use to compress {{math|''S''}} along, providing another Seifert surface of reduced complexity.  Hence, there are incompressible Seifert surfaces.

Every Seifert surface of a link is related to one another through compressions in the sense that the [[equivalence relation]] generated by compression has one equivalence class.  The inverse of a compression is sometimes called '''embedded arc surgery''' (an embedded 0-surgery).

The [[Seifert surface#Genus of a knot|genus of a link]] is the minimal [[Genus (mathematics)|genus]] of all Seifert surfaces of a link.  A Seifert surface of minimal genus is incompressible.  However, it is not in general the case that an incompressible Seifert surface is of minimal genus, so {{math|''&pi;''<sub>1</sub>}} alone cannot certify the genus of a link. [[David Gabai]] proved in particular that a genus-minimizing Seifert surface is a leaf of some taut, transversely oriented [[foliation]] of the knot complement, which can be certified with a taut [[sutured manifold hierarchy]].

Given an incompressible Seifert surface {{math|''S''}}' for a knot {{math|''K''}}, then the [[fundamental group]] of {{math|''S''<sup>3</sup> &minus; ''N''(''K'')}} splits as an [[HNN extension]] over {{math|''&pi;''<sub>1</sub>(''S'')}}, which is a [[free group]].  The two maps from {{math|''&pi;''<sub>1</sub>(''S'')}} into {{math|''&pi;''<sub>1</sub>(''S''<sup>3</sup> &minus; ''N''(''S''))}} given by pushing loops off the surface to the positive or negative side of {{math|''N''(''S'')}} are both injections.