Knot and link groups
A class of groups isomorphic to the fundamental groups (cf. Fundamental group) of the complementary spaces
of links (cf. Link)
of codimension two in the sphere
.
For the cases the groups
of smooth links of multiplicity
are distinguished by the following properties [3]: 1)
is generated as a normal subgroup by
elements; 2) the
-dimensional homology group
of
with integer coefficients and trivial action of
on
is
; and 3) the quotient group of
by its commutator subgroup
is a free Abelian group
of rank
. If
is the group of the link
, then 1) holds because
becomes the trivial group after setting the meridian equal to 1 (see below), property 2) follows from Hopf's theorem, according to which
is a quotient group of
, equal to
by Alexander duality; property 3) follows from the fact that
and
by Alexander duality.
In the case or
, necessary and sufficient conditions have not yet been found (1984). If
, then
does not split if and only if
is aspherical, i.e. is an Eilenberg–MacLane space of type
. A link
splits if and only if the group
has a presentation with deficiency larger than one [3]. The complement of a higher-dimensional
link having more than one component is never aspherical, and the complement of a higher-dimensional knot can be aspherical only under the condition
[5]. Furthermore, for
every
-dimensional knot with aspherical complement is trivial. It is also known that for
a link is trivial if and only if its group is free [3]. Suppose now that
. To obtain a presentation of the group
by a general rule (cf. Fundamental group) in
one forms a two-dimensional complex
containing the initial knot and such that
. Then the
-chains of
give a system of generators for
and going around the
-chains in
gives the relations. If one takes a cone over
for
, emanating from a point below the plane of projection, one obtains the upper Wirtinger presentation (cf. Knot and link diagrams). If for
one takes the union of the black and white surfaces obtained from the diagram of
(removing the exterior domain), one obtains the Dehn presentation.
The specification of in the form of a closed braid (cf. Braid theory; Knot and link diagrams) leads to a presentation of
in the form
, where
is a word over the alphabet
, and
in the free group
. In addition, every presentation of this type is obtained from a closed braid. For other presentations see [1], [2], [4], [7], [8]. Comparison of the upper and lower Wirtinger presentations leads to a particular kind of duality in
(cf. [7]). This may be formulated in terms of a Fox calculus:
has two presentations
and
such that for a certain equivalence
one has
and
, where the equations are taken modulo the kernel of the homomorphism of the group ring of the free group onto the group ring of
. This duality implies the symmetry of the Alexander invariant (cf. Alexander invariants).
The identity problem has been solved only for isolated classes of knots (e.g. torus and some pretzel-like knots, cf. [6], etc.). There is no algorithm (cf. [1]) for recognizing the groups of -dimensional knots from their presentation. Stronger invariants for
are the group systems
consisting of
and systems
of classes of conjugate subgroups. A subgroup
in
is called a peripheral subgroup of the component
; it is the image under the imbedding homomorphism of the fundamental group
the boundary of which is a regular neighbourhood
of the component
. If
is not the trivial knot, separated from the other components of the
-sphere, then
. The meridian and the parallel in
generate in
two elements which are also called the meridian
and the parallel
for
in the group system. In the case
the parallel is uniquely determined for the group
itself in the subgroup
, but the meridian is only determined up to a factor of the form
. For
as an invariant see Knot theory. The automorphism group of the group
has been completely studied only for torus links, for Listing knots (cf. Listing knot) and, to a higher degree, for Neuwirth knots (cf. Neuwirth knot, [2]). The representation of
in different groups, especially with regard to
, is a powerful means of distinguishing knots. E.g., the representation in the group of motions of the Lobachevskii plane allows one to describe the non-invertible knots. Metacyclic representations have been studied systematically.
If does not split, then for a subgroup
of
a space of type
is used as a covering of
which, like
, has the homotopy type of a
-dimensional complex. It follows that an Abelian subgroup of
is isomorphic to
or
; in particular,
contains no non-trivial elements of finite order. For
the peripheral subgroups
are maximal in the set of Abelian subgroups. Only the group of a toroidal link can have a centre [10]. A fundamental role is played by the subgroup
containing the elements of
whose link coefficients with the union of the oriented components
are
. If
, then
is the commutator subgroup; generally
. Therefore
may be taken as group of a covering
over
with infinite cyclic group
of covering transformations. If
is a connected oriented surface in
with boundary
, then it is covered in
by a countable system of surfaces
, which decompose
into a countable number of pieces
(where
). Hence one obtains that
is the limit of the diagram
![]() |
where all the ,
are induced inclusions. It turns out that either they are all isomorphisms or no two are epimorphisms [2]. If the genus of a connected
is equal to the genus
of its link (such a
is called completely non-split), then all the
,
are monomorphisms and
is either a free group of rank
or is not finitely generated (and not free, if the reduced Alexander polynomial is not zero; this is so for knots, in particular). A completely non-split link with finitely generated
is called a Neuwirth link.
References
[1] | R.H. Crowell, R.H. Fox, "Introduction to knot theory" , Ginn (1963) |
[2] | L.P. Neuwirth, "Knot groups" , Princeton Univ. Press (1965) |
[3] | J.A. Hillman, "Alexander ideals of links" , Springer (1981) |
[4] | C.McA. Gordon, "Some aspects of clasical knot theory" , Knot theory. Proc. Sem. Plans-sur-Bex, 1977 , Lect. notes in math. , 685 , Springer (1978) pp. 1–60 |
[5] | B. Eckmann, "Aspherical manifolds and higher-dimensional knots" Comm. Math. Helv. , 51 (1976) pp. 93–98 |
[6] | K. Reidemeister, "Ueber Knotengruppen" Abh. Math. Sem. Univ. Hamburg , 6 (1928) pp. 56–64 |
[7] | G. Hotz, "Arkandenfadendarstellung von Knoten und eine neue Darstellung der Knotengruppe" Abh. Math. Sem. Univ. Hamburg , 24 (1960) pp. 132–148 |
[8] | H.F. Trotter, "Homology of group systems with applications to knot theory" Ann. of Math. , 76 (1962) pp. 464–498 |
[9] | H.F. Trotter, "Non-invertible knots exist" Topology , 2 (1964) pp. 275–280 |
[10] | G. Burde, H. Zieschang, "Eine Kennzeichnung der Torusknotten" Math. Ann. , 167 (1966) pp. 169–176 |
Comments
An -link
is splittable if there is an
-sphere
such that
meets each of the two components of
.
The deficiency of a presentation of a group by means of generators and relations
is
, [a1].
References
[a1] | P.E. Schapp, "Combinatorial group theory" , Springer (1977) pp. Chapt. II, Sect. 2 |
[a2] | L.H. Kauffman, "On knots" , Princeton Univ. Press (1987) |
[a3] | J.S. Birman, "Braids, links and mapping class groups" , Princeton Univ. Press (1974) |
[a4] | D. Rolfsen, "Knots and links" , Publish or Perish (1976) |
Knot and link groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Knot_and_link_groups&oldid=47506