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)