##### Actions

A class of groups isomorphic to the fundamental groups (cf. Fundamental group) $G ( K) = \pi _ {1} ( M ( k) )$ of the complementary spaces $M ( k) = S ^ {n} \setminus k$ of links (cf. 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 [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; 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 of type $K ( G , 1)$. A link $k$ splits if and only if the group $G$ has a presentation with deficiency larger than one [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 $G \approx \mathbf Z$[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 [3]. Suppose now that $n = 3$. To obtain a presentation of the group $G ( k)$ by a general rule (cf. 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). 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; 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 [1], [2], [4], [7], [8]. Comparison of the upper and lower Wirtinger presentations leads to a particular kind of duality in $G ( k)$( cf. [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).

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 $3$- 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. The automorphism group of the group $G$ 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 $G$ 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 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 [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 [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

 [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

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 $n - m$, [a1].