algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

, a peripheral subgroup for a space-subspace pair ''X'' ⊃ ''Y'' is a certain subgroup of the

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

of the complementary space, π₁(''X'' − ''Y''). Its

conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wor ...

is an invariant of the pair (''X'',''Y''). That is, any

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...

(''X'', ''Y'') → (''X''′, ''Y''′) induces an

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

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 In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...

as a complete

algebraic invariant Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descrip ...

of knots. There is a systematic way to choose generators for a peripheral subgroup of a knot in

3-space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...

, 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 In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a ...

Full definition

Let ''Y'' be a subspace of the

path-connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that ...

topological space ''X'', whose complement ''X'' − ''Y'' is path-connected. Fix a basepoint ''x'' ∈ ''X'' − ''Y''. For each path component ''V''_''i'' of ''X'' − ''Y''∩''Y'', choose a

path A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desire p ...

γ_i from ''x'' to a point in ''V''_''i''. An element �nbsp;∈ π₁(''X'' − ''Y'', ''x'') is called peripheral with respect to this choice if it is represented by a loop in ''U'' ∪ ∪ _''i''γ_''i'' for every neighborhood ''U'' of ''Y''. The set of all peripheral elements with respect to a given choice forms a subgroup of π₁(''X'' − ''Y'', ''x''), called a peripheral subgroup. 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''. Then it would move around through ''U'' however it likes (avoiding ''Y''). Finally it would return to the basepoint ''x'' via γ. Since ''U'' can be a very tight envelope around ''Y'', the loop has to stay close to ''Y''. Any two peripheral subgroups of π₁(''X'' − ''Y'', ''x''), resulting from different choices of paths γ_i, are conjugate in π₁(''X'' − ''Y'', ''x''). Also, every conjugate of a peripheral subgroup is itself peripheral with respect to some choice of paths γ_i. Thus the peripheral subgroup's

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.

In knot theory

The peripheral subgroups for a

tame knot Tame may refer to: *Taming, the act of training wild animals *River Tame, Greater Manchester *River Tame, West Midlands and the Tame Valley *Tame, Arauca, a Colombian town and municipality * "Tame" (song), a song by the Pixies from their 1989 alb ...

''K'' in R³ are isomorphic to Z ⊕ Z if the knot is nontrivial, Z if it is the

unknot In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embe ...

. 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 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. 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 ambient isotopy, 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'']^''n'' and [''m'']⁻¹, respectively (''n'' ∈ 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 (m′,l′,n) is a orientation (vector space), 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 (geometry), normal to the tube at ''y''. (Assume that representatives ''l'' and ''m'' are chosen to be smooth function, 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]. Granny knot and square knot

Example: Square knot versus granny knot

The square knot (mathematics), square knot and the granny knot (mathematics), granny knot are distinct knots, and have non-homeomorphic knot complement, 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 two trefoil knots

Example: Trefoil versus mirror trefoil

The trefoil and its chiral knot, 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 R³ 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.

Wirtinger presentation

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.

References

* Ralph Fox, Fox, Ralph H.,
A quick trip through knot theory
', in: M.K. Fort (Ed.), "Topology of 3-Manifolds and Related Topics", Prentice-Hall, NJ, 1961, pp. 120–167. *{{Citation , last1=Waldhausen , first1=Friedhelm , author1-link=Friedhelm Waldhausen , title=On irreducible 3-manifolds which are sufficiently large , jstor=1970594 , mr=0224099 , year=1968 , journal=Annals of Mathematics , series=Second Series , issn=0003-486X , volume=87 , issue=1 , pages=56–88 , doi=10.2307/1970594, url=https://pub.uni-bielefeld.de/record/1782185 * Max Dehn, Dehn, Max,
Die beiden Kleeblattschlingen
', ''Mathematische Annalen'' 75 (1914), no. 3, 402–413. Algebraic topology Homotopy theory Knot theory