Peripheral subgroup

An element [α] ∈ π1(X − Y, x) is called peripheral with respect to this choice if it is represented by a loop in U ∪  ∪ iγi for every neighborhood U of Y.

In the diagram, a peripheral loop would start at the basepoint x and travel down the path γ until it's inside the neighborhood U of the subspace Y.

Any two peripheral subgroups of π1(X − Y, x), resulting from different choices of paths γi, are conjugate in π1(X − Y, x).

A longitude is a loop that runs from the basepoint x along a path γ 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 γ.

(The property of being a longitude or meridian is well-defined because the tubular neighborhoods of a tame knot are all ambiently isotopic.)

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 (m′,l′,n) is a positively oriented basis for R3, where m′ is the tangent vector of m based at y, l′ is the tangent vector of l based at y, and n is an outward-pointing normal to the tube at y.

If so chosen, the peripheral system is a complete invariant for knots, as proven in [Waldhausen 1968].

The trefoil and its mirror image are distinct knots, and consequently there is no orientation-preserving homeomorphism between their complements.

Nonetheless, it was shown in [Dehn 1914] that no isomorphism of these knot groups preserves the peripheral system selected as described above.