Knot and link groups

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

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 $n - m$, [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)