View source for Knot and link groups

<!--
k0555801.png
$#A+1 = 149 n = 0
$#C+1 = 149 : ~/encyclopedia/old_files/data/K055/K.0505580 Knot and link groups
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A class of groups isomorphic to the fundamental groups (cf. [[Fundamental group|Fundamental group]])  $  G ( K) = \pi _ {1} ( M ( k) ) $
of the complementary spaces  $  M ( k) = S  ^ {n} \setminus  k $
of links (cf. [[Link|Link]])  $  k $
of codimension two in the sphere  $  S  ^ {n} $.

For the cases  $  n \geq  5 $
the groups  $  G $
of smooth links of multiplicity  $  \mu $
are distinguished by the following properties [[#References|[3]]]: 1)  $  G $
is generated as a normal subgroup by  $  \mu $
elements; 2) the  $  2 $-
dimensional homology group  $  H _ {2} ( G ;  \mathbf Z ) $
of  $  G $
with integer coefficients and trivial action of  $  G $
on  $  \mathbf Z $
is  $  0 $; 
and 3) the quotient group of  $  G $
by its commutator subgroup  $  G  ^  \prime  $
is a free Abelian group  $  J  ^  \mu  $
of rank  $  \mu $. 
If  $  G $
is the group of the link  $  k $, 
then 1) holds because  $  G $
becomes the trivial group after setting the meridian equal to 1 (see below), property 2) follows from Hopf's theorem, according to which  $  H  ^ {2} ( G ;  \mathbf Z ) $
is a quotient group of  $  H  ^ {2} ( M ( k) ;  \mathbf Z ) $, 
equal to  $  0 $
by [[Alexander duality|Alexander duality]]; property 3) follows from the fact that  $  G / G  ^  \prime  \approx H _ {1} ( M( k) ;  \mathbf Z ) $
and  $  H _ {1} ( M ( k) ;  \mathbf Z ) = J  ^  \mu  $
by Alexander duality.

In the case  $  n = 3 $
or  $  4 $, 
necessary and sufficient conditions have not yet been found (1984). If  $  n = 3 $, 
then  $  k $
does not split if and only if  $  M ( k) $
is aspherical, i.e. is an [[Eilenberg–MacLane space|Eilenberg–MacLane space]] of type  $  K ( G , 1) $. 
A link  $  k $
splits if and only if the group  $  G $
has a presentation with deficiency larger than one [[#References|[3]]]. The complement of a higher-dimensional  $  ( n \geq  4 ) $
link having more than one component is never aspherical, and the complement of a higher-dimensional knot can be aspherical only under the condition  $  G \approx \mathbf Z $[[#References|[5]]]. Furthermore, for  $  n \geq  6 $
every  $  n $-
dimensional knot with aspherical complement is trivial. It is also known that for  $  n = 3 $
a link is trivial if and only if its group is free [[#References|[3]]]. Suppose now that  $  n = 3 $. 
To obtain a presentation of the group  $  G ( k) $
by a general rule (cf. [[Fundamental group|Fundamental group]]) in  $  S  ^ {3} $
one forms a two-dimensional complex  $  K $
containing the initial knot and such that  $  \pi _ {1} ( S  ^ {3} - K ) = 1 $. 
Then the  $  2 $-
chains of  $  K $
give a system of generators for  $  G ( k) $
and going around the  $  1 $-
chains in  $  K \setminus  k $
gives the relations. If one takes a cone over  $  k $
for  $  K $, 
emanating from a point below the plane of projection, one obtains the upper Wirtinger presentation (cf. [[Knot and link diagrams|Knot and link diagrams]]). If for  $  K $
one takes the union of the black and white surfaces obtained from the diagram of  $  k $(
removing the exterior domain), one obtains the Dehn presentation.

The specification of  $  k $
in the form of a closed braid (cf. [[Braid theory|Braid theory]]; [[Knot and link diagrams|Knot and link diagrams]]) leads to a presentation of  $  G ( k) $
in the form  $  \{ {s _ {i} } : {s _ {i} = L _ {i} s _ {k} L _ {i}  ^ {-} 1 } \} $, 
where  $  L _ {1} $
is a word over the alphabet  $  s _ {i} , s _ {i}  ^ {-} 1 $, 
and  $  \prod _ {i=} 1  ^ {N} ( L _ {i} s _ {k} L _ {i}  ^ {-} 1 ) = \prod _ {i=} 1  ^ {N} s _ {i} $
in the free group  $  \{ s _ {i} \} $. 
In addition, every presentation of this type is obtained from a closed braid. For other presentations see [[#References|[1]]], [[#References|[2]]], [[#References|[4]]], [[#References|[7]]], [[#References|[8]]]. Comparison of the upper and lower Wirtinger presentations leads to a particular kind of duality in  $  G ( k) $(
cf. [[#References|[7]]]). This may be formulated in terms of a Fox calculus:  $  G ( k) $
has two presentations  $  ( x _ {i} ;  r _ {j} ) $
and  $  ( y _ {i} ;  s _ {j} ) $
such that for a certain equivalence  $  \theta :  ( x _ {i} ;  r _ {j} ) \rightarrow ( y _ {i} ;  s _ {j} ) $
one has  $  \theta x _ {i} \equiv y _ {i}  ^ {-} 1 $
and  $  \theta ( \partial  r _ {i} / \partial  x _ {j} ) ( x _ {j} - 1 ) \equiv ( \partial  s _ {j} / \partial  y _ {j} )( y _ {i} - 1 ) $, 
where the equations are taken modulo the kernel of the homomorphism of the group ring of the free group onto the group ring of  $  G / G  ^  \prime  $. 
This duality implies the symmetry of the Alexander invariant (cf. [[Alexander invariants|Alexander invariants]]).

The identity problem has been solved only for isolated classes of knots (e.g. torus and some pretzel-like knots, cf. [[#References|[6]]], etc.). There is no algorithm (cf. [[#References|[1]]]) for recognizing the groups of  $  3 $-
dimensional knots from their presentation. Stronger invariants for  $  k $
are the group systems  $  \langle  G , T _ {i} \rangle $
consisting of  $  G ( k) $
and systems  $  T _ {i} $
of classes of conjugate subgroups. A subgroup  $  S _ {i} $
in  $  T _ {i} $
is called a peripheral subgroup of the component  $  k _ {i} $; 
it is the image under the imbedding homomorphism of the fundamental group  $  \pi _ {1} ( \partial  N ( k _ {i} ) ) $
the boundary of which is a regular neighbourhood  $  N ( k _ {i} ) $
of the component  $  k _ {i} \subset  k $. 
If  $  k _ {i} $
is not the trivial knot, separated from the other components of the  $  2 $-
sphere, then  $  s _ {i} \approx \pi _ {1} ( \partial  N ( k _ {i} ) ) $. 
The meridian and the parallel in  $  \partial  N ( k _ {i} ) $
generate in  $  S _ {i} $
two elements which are also called the meridian  $  m _ {i} $
and the parallel  $  l _ {i} $
for  $  k _ {i} $
in the group system. In the case  $  \mu = 1 $
the parallel is uniquely determined for the group  $  G $
itself in the subgroup  $  S _ {i} $, 
but the meridian is only determined up to a factor of the form  $  l _ {i}  ^ {n} $. 
For  $  \langle  G _ {i} , T _ {i} \rangle $
as an invariant see [[Knot theory|Knot theory]]. The automorphism group of the group  $  G $
has been completely studied only for torus links, for Listing knots (cf. [[Listing knot|Listing knot]]) and, to a higher degree, for Neuwirth knots (cf. [[Neuwirth knot|Neuwirth knot]], [[#References|[2]]]). The representation of  $  G $
in different groups, especially with regard to  $  \langle  G , T _ {i} \rangle $, 
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  $  k $
does not split, then for a subgroup  $  F $
of  $  G ( k) $
a space of type  $  K ( F ;  1 ) $
is used as a covering of  $  M $
which, like  $  M $, 
has the [[Homotopy type|homotopy type]] of a  $  2 $-
dimensional complex. It follows that an Abelian subgroup of  $  G ( k) $
is isomorphic to  $  J $
or  $  J \oplus J $; 
in particular,  $  G ( k) $
contains no non-trivial elements of finite order. For  $  \mu = 1 $
the peripheral subgroups  $  S _ {i} $
are maximal in the set of Abelian subgroups. Only the group of a toroidal link can have a centre [[#References|[10]]]. A fundamental role is played by the subgroup  $  L ( k) $
containing the elements of  $  G ( k) $
whose link coefficients with the union of the oriented components  $  k _ {i} $
are  $  0 $. 
If  $  \mu = 1 $, 
then  $  L ( k) $
is the commutator subgroup; generally  $  G ( k) / L ( k) \approx J $. 
Therefore  $  L ( k) $
may be taken as group of a covering  $  \widetilde{M}  _ {0} $
over  $  M ( k) $
with infinite cyclic group  $  J $
of covering transformations. If  $  F ( k) $
is a connected oriented surface in  $  S  ^ {3} $
with boundary  $  k $, 
then it is covered in  $  \widetilde{M}  _ {0} $
by a countable system of surfaces  $  \widetilde{F}  _ {j} $, 
which decompose  $  \widetilde{M}  _ {0} $
into a countable number of pieces  $  M _ {j} $(
where  $  \partial  M _ {j} = F _ {j} \cup F _ {j+} 1 $). 
Hence one obtains that  $  L ( k) $
is the limit of the diagram

$$ 
\dots \leftarrow ^ { {i _ {j}  * } } \ 
\pi _ {1} F _ {j} \ 
\rightarrow ^ { {i _ {j}  * } } \ 
\pi _ {1} M _ {j} \ 
\leftarrow ^ { {i _ {j}  + 1 * }  ^  \prime  } \ 
\pi _ {1} F _ {j+} 1 \ 
\leftarrow ^ { {i _ {j}  + 1 * } } \dots ,
$$

where all the  $  i _ {j * }  $, 
$  i _ {j * }   ^  \prime  $
are induced inclusions. It turns out that either they are all isomorphisms or no two are epimorphisms [[#References|[2]]]. If the genus of a connected  $  F ( k) $
is equal to the genus  $  \gamma ( k) $
of its link (such a  $  k $
is called completely non-split), then all the  $  i _ {j * }  $, 
$  i _ {j * }   ^  \prime  $
are monomorphisms and  $  L ( k) $
is either a free group of rank  $  2 \gamma + \mu - 1 $
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  $  L ( k) $
is called a Neuwirth link.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.H. Crowell,   R.H. Fox,   "Introduction to knot theory" , Ginn  (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.P. Neuwirth,   "Knot groups" , Princeton Univ. Press  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Hillman,   "Alexander ideals of links" , Springer  (1981)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  B. Eckmann,   "Aspherical manifolds and higher-dimensional knots"  ''Comm. Math. Helv.'' , '''51'''  (1976)  pp. 93–98</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K. Reidemeister,   "Ueber Knotengruppen"  ''Abh. Math. Sem. Univ. Hamburg'' , '''6'''  (1928)  pp. 56–64</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  G. Hotz,   "Arkandenfadendarstellung von Knoten und eine neue Darstellung der Knotengruppe"  ''Abh. Math. Sem. Univ. Hamburg'' , '''24'''  (1960)  pp. 132–148</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  H.F. Trotter,   "Homology of group systems with applications to knot theory"  ''Ann. of Math.'' , '''76'''  (1962)  pp. 464–498</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  H.F. Trotter,   "Non-invertible knots exist"  ''Topology'' , '''2'''  (1964)  pp. 275–280</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  G. Burde,   H. Zieschang,   "Eine Kennzeichnung der Torusknotten"  ''Math. Ann.'' , '''167'''  (1966)  pp. 169–176</TD></TR></table>

====Comments====
An $n$-link $L \subset S^{n+2}$
is splittable if there is an $(n + 1)$-sphere  $S  ^ {n+ 1} \subseteq S  ^ {n+ 2} \setminus  L $
such that $L$ meets each of the two components of  $  S  ^ {n+ 2} \setminus  S^{n+1}$.

The deficiency of a presentation of a group by means of generators $x_{1} \dots x_{n}$
and relations $r_{1} \dots r_{m}$ is $n - m$, [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.E. Schapp,   "Combinatorial group theory" , Springer  (1977)  pp. Chapt. II, Sect. 2</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.H. Kauffman,   "On knots" , Princeton Univ. Press  (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.S. Birman,   "Braids, links and mapping class groups" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D. Rolfsen,   "Knots and links" , Publish or Perish  (1976)</TD></TR></table>