In algebraic topology, a peripheral subgroup for a space-subspace pair X â Y is a certain subgroup of the fundamental group of the complementary space, ÃÂ<sub>1</sub>(X â Y). Its conjugacy class is an invariant of the pair (X,Y). That is, any homeomorphism (X, Y) â (Xâ², Yâ²) induces an isomorphism ÃÂ<sub>1</sub>(X â Y) â ÃÂ<sub>1</sub>(Xâ² â Yâ²) taking peripheral subgroups to peripheral subgroups.
A peripheral subgroup consists of loops in X â 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.
Let Y be a subspace of the path-connected topological space X, whose complement X â Y is path-connected. Fix a basepoint x â X â Y. For each path component V<sub>i</sub> of <span style="text-decoration: overline">X â Y</span>â©<span style="text-decoration: overline">Y</span>, choose a path ó<sub>i</sub> from x to a point in V<sub>i</sub>. An element [ñ] â ÃÂ<sub>1</sub>(X â Y, x) is called peripheral with respect to this choice if it is represented by a loop in U ⪠<big> ⪠</big><sub>i</sub>ó<sub>i</sub> for every neighborhood U of Y. The set of all peripheral elements with respect to a given choice forms a subgroup of ÃÂ<sub>1</sub>(X â Y, x), called a peripheral subgroup.
In the diagram, a peripheral loop would start at the basepoint x and travel down the path ó until it's inside the neighborhood U of the subspace Y. Then it would move around through U however it likes (avoiding Y). Finally it would return to the basepoint x via ó. Since U can be a very tight envelope around Y, the loop has to stay close to Y.
Any two peripheral subgroups of ÃÂ<sub>1</sub>(X â Y, x), resulting from different choices of paths ó<sub>i</sub>, are conjugate in ÃÂ<sub>1</sub>(X â Y, x). Also, every conjugate of a peripheral subgroup is itself peripheral with respect to some choice of paths ó<sub>i</sub>. Thus the peripheral subgroup's conjugacy class is an invariant of the pair (X, Y).
A peripheral subgroup, together with an ordered set of generators, is called a peripheral system for the pair (X, Y). If a systematic method is specified for selecting these generators, the peripheral system is, in general, a stronger invariant than the peripheral subgroup alone. In fact, it is a complete invariant for knots.
The peripheral subgroups for a tame knot K in R<sup>3</sup> are isomorphic to Z â Z if the knot is nontrivial, Z if it is the unknot. They are generated by two elements, called a longitude [l] and a meridian [m]. (If K is the unknot, then [l] is a power of [m], and a peripheral subgroup is generated by [m] alone.) A longitude is a loop that runs from the basepoint x along a path ó to a point y on the boundary of a tubular neighborhood of K, then follows along the tube, making one full lap to return to y, then returns to x via ó. A meridian is a loop that runs from x to y, then circles around the tube, returns to y, then returns to x. (The property of being a longitude or meridian is well-defined because the tubular neighborhoods of a tame knot are all ambiently isotopic.) Note that every knot group has a longitude and meridian; if [l] and [m] are a longitude and meridian in a given peripheral subgroup, then so are [l]÷[m]<sup>n</sup> and [m]<sup>−1</sup>, respectively (n â Z). In fact, these are the only longitudes and meridians in the subgroup, and any pair will generate the subgroup.
A peripheral system for a knot can be selected by choosing generators [l] and [m] such that the longitude l has linking number 0 with K, and the ordered triple (mâ²,lâ²,n) is a positively oriented basis for R<sup>3</sup>, where mâ² is the tangent vector of m based at y, lâ² is the tangent vector of l based at y, and n is an outward-pointing normal to the tube at y. (Assume that representatives l and m are chosen to be smooth on the tube and cross only at y.) If so chosen, the peripheral system is a complete invariant for knots, as proven in [Waldhausen 1968].
The square knot and the granny knot are distinct knots, and have non-homeomorphic complements. However, their knot groups are isomorphic. Nonetheless, it was shown in [Fox 1961] that no isomorphism of their knot groups carries a peripheral subgroup of one to a peripheral subgroup of the other. Thus the peripheral subgroup is sufficient to distinguish these knots.
The trefoil and its mirror image are distinct knots, and consequently there is no orientation-preserving homeomorphism between their complements. However, there is an orientation-reversing self-homeomorphism of R<sup>3</sup> that carries the trefoil to its mirror image. This homeomorphism induces an isomorphism of the knot groups, carrying a peripheral subgroup to a peripheral subgroup, a longitude to a longitude, and a meridian to a meridian. Thus the peripheral subgroup is not sufficient to distinguish these knots. Nonetheless, it was shown in [Dehn 1914] that no isomorphism of these knot groups preserves the peripheral system selected as described above. An isomorphism will, at best, carry one generator to a generator going the "wrong way". Thus the peripheral system can distinguish these knots.
It is possible to express longitudes and meridians of a knot as words in the Wirtinger presentation of the knot group, without reference to the knot itself.