Homotopy theory | Knot theory | Algebraic topology
Peripheral subgroup
In algebraic topology, a peripheral subgroup for a space-subspace pair X ⊃ Y is a certain subgroup of the fundamental group of the complementary space, π1(X − Y). Its conjugacy class is an invariant of the pair (X,Y). That is, any homeomorphism (X, Y) → (X′, Y′) induces an isomorphism π1(X − Y) → π1(X′ − Y′) taking peripheral subgroups to peripheral subgroups. A peripheral subgroup consists of loops in X − Y which are peripheral to Y, that is, which stay "close to" Y (except when passing to and from the basepoint). When an ordered set of generators for a peripheral subgroup is specified, the subgroup and generators are collectively called a peripheral system for the pair (X, Y). Peripheral systems are used in knot theory as a complete algebraic invariant of knots. There is a systematic way to choose generators for a peripheral subgroup of a knot in 3-space, such that distinct knot types always have algebraically distinct peripheral systems. The generators in this situation are called a longitude and a meridian of the knot complement. (Wikipedia).
