In
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, π
1(''X'' − ''Y''). Its
conjugacy class 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 ...
Ï€
1(''X'' − ''Y'') → π
1(''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 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;∈ Ï€
1(''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 π
1(''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 π
1(''X'' − ''Y'', ''x''), resulting from different choices of paths γ
i, are
conjugate in π
1(''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
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.
In knot theory
The peripheral subgroups for a
tame knot ''K'' in R
3 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
In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.
The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the pla ...
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
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'' and
'm''sup>−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
positively oriented basis for R
3, where m′ is the
tangent vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...
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''. (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
aldhausen 1968
Example: Square knot versus granny knot
The
square knot and the
granny knot
The granny knot is a binding knot, used to secure a rope or line around an object. It is considered inferior to the reef knot (square knot), which it superficially resembles. Neither of these knots should be used as a bend knot for attaching tw ...
are distinct knots, and have non-homeomorphic
complements. However, their
knot groups are isomorphic. Nonetheless, it was shown in
ox 1961that 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.
Example: Trefoil versus mirror trefoil
The
trefoil and its
mirror image
A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off from substances ...
are distinct knots, and consequently there is no orientation-preserving homeomorphism between their complements. However, there is an orientation-reversing self-homeomorphism of R
3 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
ehn 1914that 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
*
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
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
History
The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the ...
, 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
*
Dehn, Max,
Die beiden Kleeblattschlingen', ''
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...
'' 75 (1914), no. 3, 402–413.
Algebraic topology
Homotopy theory
Knot theory