by compressing S along D. There is a tubular neighborhood of D whose closure is an embedding of D × [-1,1] with D × 0 being identified with D and with Then is a new properly embedded surface obtained by compressing S along D. A non-negative complexity measure on compact surfaces without 2-sphere components is b0(S) − χ(S), where b0(S) is the zeroth Betti number (the number of connected components) and χ(S) is the Euler characteristic. When compressing a compressible surface along a nontrivial compressing disk, the Euler characteristic increases by two, while b0 might remain the same or increase by 1. Thus, every properly embedded compact surface without 2-sphere components is related to an incompressible surface through a sequence of compressions. Sometimes we drop the condition that S be compressible. If D were to bound a disk inside S (which is always the case if S is incompressible, for example), then compressing S along D would result in a disjoint union of a sphere and a surface homeomorphic to S. The resulting surface with the sphere deleted might or might not be isotopic to S, and it will be if S is incompressible and M is irreducible. Algebraically incompressible surfaces There is also an algebraic version of incompressibility. Suppose is a proper embedding of a compact surface in a 3-manifold. Then S is π1-injective (or algebraically incompressible) if the induced map on fundamental groups is injective. In general, every π1-injective surface is incompressible, but the reverse implication is not always true. For instance, the Lens space L(4,1) contains an incompressible Klein bottle that is not π1-injective. However, if S is two-sided, the loop theorem implies Kneser's lemma, that if S is incompressible, then it is π1-injective. Seifert surfaces A Seifert surface S for an oriented link L is an oriented surface whose boundary is L with the same induced orientation. If S is not π1 injective in S3 − N(L), where N(L) is a tubular neighborhood of L, then the loop theorem gives a compressing disk that one may use to compress 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