genus of a link is the minimal 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 π1 alone cannot certify the genus of a link. 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 S for a knot K, then the fundamental group of S3 − N(K) splits as an HNN extension over π1(S), which is a free group. The two maps from π1(S) into π1(S3 − N(S)) given by pushing loops off the surface to the positive or negative side of N(S) are both injections. See also Haken manifold Virtually Haken conjecture Thurston norm Boundary-incompressible surface References W. Jaco, Lectures on Three-Manifold Topology, volume 43 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, R.I., 1980. http://users.monash.edu/~jpurcell/book/08Essential.pdf https://homepages.warwick.ac.uk/~masgar/Articles/Lackenby/thrmans3.pdf D. Gabai, "Foliations and the topology of 3-manifolds." Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 1, 77–80. 3-manifolds