H. Tzishang

On a Problem of Neuwirth Concerning Knot Groups

(Presented by Academician P. S. Aleksandrov on 17 VII 1963)

In the book \((^1)\), R. H. Fox proposed several problems on knots, among them the following Problem No. 3 (first posed by Neuwirth \((^2)\)): Is it true that every knot group can be represented as the free product of two free groups of the same rank \(n\) with amalgamated subgroup of rank \(2n-1\)?

Neuwirth \((^2)\) proved this assertion for all alternating knots and for knots with at most 9 crossings except \(9_{46}\) (in the standard notation \((^3)\)). Here we shall show that the assertion is also true for \(9_{46}\). Moreover, we shall strengthen it and prove that, in a certain sense, the converse assertion holds, namely, that every group having a representation of the strengthened form is the group of a knot in some three-dimensional manifold.

1. Let \(V\) be an oriented handlebody of genus \(n\), and let \(k\) be a simple closed path on the surface. We shall say that \(k\) has full rank if its homotopy class in \(V\) cannot be represented by free generators fewer in number than \(n\). If \(k\) does not cut the surface, then \(k\) can be completed to a canonical cut \(k_1, k_2 = k^{-1}, k_3, \ldots, k_{2n}\) of the surface (i.e. \(k_1, \ldots, k_{2n}\) are simple closed paths which cut the surface into a disk with boundary

\[ \prod_{i=1}^{n} b_{2i-1} b_{2i} b_{2i-1}^{-1} b_{2i}^{-1}. \]

By \(\mathfrak F\) we denote the fundamental group of the handlebody \(V\) with base point at the initial point of the path \(k\). Let \(\mathfrak u_k\) be the subgroup of the group \(\mathfrak F\) consisting of those classes which contain a path with the following properties: it lies on the surface, its intersection with \(k\) consists only of the initial point, where it only touches \(k\). In addition, the path leaves \(k\) on one definite side. The following proposition holds:

In order that \(k\) have full rank, it is necessary and sufficient that \(\mathfrak u_k\) have rank \(2n-1\).

Here we shall need only the sufficiency, which follows from the “loop theorem” of Papakyriakopoulos \((^4)\) and Dehn’s lemma.

Let \(M\) be a three-dimensional manifold and \((V, W)\) a Heegaard splitting for \(M\). Let a simple closed path \(k\), not cutting \(F\), be situated on the common surface \(F\) of the handlebodies \(V\) and \(W\). We obtain the group of the knot \(k\) in the following way: complete \(k\) to a canonical cut \(k_1, k_2 = k^{-1}, \ldots, k_{2n}\) of the surface and choose free generators \(S_1, \ldots, S_n\) of the fundamental group of the handlebody \(V\) and \(T_1, \ldots, T_n\) for \(W\). Let \(K_1, \ldots, K_{2n}\) be the words for the paths \(k_1, \ldots, k_{2n}\) in the generators \(S_1, \ldots, S_n\), and let \(H_1, \ldots, H_{2n}\) be the corresponding words in \(T_1, \ldots, T_n\). From van Kampen’s theorem \((^5)\) it follows that the group \(\mathfrak G\) of the knot \(k\) has generators \(S_1, \ldots, S_n, T_1, \ldots, T_n\) and defining relations \(K_2 = H_2, \ldots, K_{2n} = H_{2n}\). If \(k\) has full rank both in \(V\) and in \(W\), then its group is the free product of two free groups of rank \(n\) with amalgamated subgroup of rank \(2n-1\).

More precisely, the following holds:

A. \(\mathfrak G\) contains two free subgroups \(\mathfrak F_1\) and \(\mathfrak F_2\) with free generators \(S_1, \ldots, S_n\) and \(T_1, \ldots, T_n\). There exist in \(\mathfrak F_1\) (and in \(\mathfrak F_2\)) elements \(K_1, \ldots, K_{2n}\) (or \(H_1, \ldots, H_{2n}\)) which generate \(\mathfrak F_1\) (or \(\mathfrak F_2\)) and satisfy the equality

\[ \prod_{i=1}^{n} [K_{2i-1}, K_{2i}] = 1 \quad \left(\text{or } \prod_{i=1}^{n} [H_{2i-1}, H_{2i}] = 1\right). \]
\(K_2 = H_2, \ldots, K_{2n} = H_{2n}\) are defining relations of the group \(\mathfrak G\).

2. In the case of the knot \(9_{46}\) we make the isotopic deformations shown in Fig. 1. In Fig. 1e we see that \(9_{46}\) has the word \(S_1 S_2^{-1} S_1^{-1} S_2\) on \(V\) and \(T_1^{-1} T_2 T_1^{-1} T_2^{-1} T_1^{2} T_2^{-1} T_1 T_2\) on \(W\). Since both words have no period and both are not contained in a system of free generators, \(9_{46}\) has full rank on both sides.

Fig. 1. \(a\)—\(9_{46}\) in the standard projection \((^3)\); \(b\)—\(d\)—deformed knots; \(e\)—\(V\) and \(W\) denote full handlebodies, on their common surface lies the knot; the disks \(S_1\) and \(S_2\) form a cut for \(V\); \(T_1\) and \(T_2\)—a cut for \(W\)

In the ordinary projection (i.e. Fig. 1a) the natural Heegaard splitting for \(9_{46}\) has genus 5; and therefore the relations obtained from it are not suitable for Neuwirth’s assertion.

3. If, conversely, the group \(\mathfrak G\) has form A, then it is the group of a knot lying on a Heegaard splitting of some manifold and having full rank on both sides.

We take two full handlebodies \(V\) and \(W\) of genus \(n\) with surfaces \(F_1\) and \(F_2\), and choose arbitrary systems of free generators \(S_1, \ldots, S_n\), or \(T_1, \ldots, T_n\), for the fundamental groups of the handlebodies \(V\) and \(W\). Then there exist such canonical cuts \(k_1, \ldots, k_{2n}\) and \(h_1, \ldots, h_{2n}\) of the surfaces \(F_1\) or \(F_2\) that their paths induce, in the corresponding fundamental groups, the words \(K_1, \ldots, K_{2n}\) or \(H_1, \ldots, H_{2n}\) \((^6)\). We identify \(F_1\) and \(F_2\) by means of a homeomorphism that maps \(k_i\) to \(h_i\), and we obtain a three-dimensional manifold \(M\). We regard \(k_2\) as a knot in \(M\). In this case \(\mathfrak G\) is its group.

Let \(k\) and \(k'\) be two knots whose groups have presentations of type A, obtained from canonical cuts of surfaces for Heegaard splittings. If there exists an automorphism \(\alpha: \mathfrak G \to \mathfrak G''\) with \(\alpha K_i = K_i'\) and \(\alpha H_1 = H_1'^{*}\), then there exists a homeomorphism \(\eta\) between the manifolds containing \(k\) or \(k'\), which carries \(k\) to \(k'\). \(\eta\) is constructed as follows. \(k\) and \(k'\) lie on the surfaces of Heegaard splittings \((V, W)\) or \((V', W')\). First we construct

* By primes we denote objects for \(k'\).

homeomorphism between the surfaces which carries the cut defining the presentation \(A\) for \(k\) into the cut corresponding to \(k'\). This homeomorphism can be extended to homeomorphisms between \(V\) and \(V'\) and between \(W\) and \(W'\) \((^{6})\).

From the presentation \(A\) for the knot group \(k\) one easily obtains the subgroup induced by the fundamental group of the surface of a regular neighborhood of \(k\): it is generated by the elements \(H_1^{-1}K_1\) and \(K_2\). Sometimes our conditions concerning the position of the knot on a Heegaard splitting surface are superfluous, as follows from Neuwirth’s work \((^{7})\). In this case, solely from knowledge of the generators and relations for the knot group, one can obtain that it lies on a Heegaard splitting surface having full rank on both sides.

Moscow State University
named after M. V. Lomonosov

Received
7 VI 1963

REFERENCES

\({}^{1}\) R. H. Fort, jr. Topology of 3-Manifolds, N. Y., 1962, p. 168.
\({}^{2}\) L. Neuwirth, Knot Groups, Princeton Thesis, 1959.
\({}^{3}\) K. Reidemeister, Knotentheorie, Ergebn. Math., 1, Berlin, 1932.
\({}^{4}\) C. D. Papakyriakopoulos, Proc. London Math. Soc., 3, Ser. 7, 281 (1957).
\({}^{5}\) E. R. van Kampen, Ann. J. Math., 55, 261 (1933).
\({}^{6}\) H. Zieschang, Abh. math. Sem. Univ. Hamburg, 27, H. 1/2 (1964).
\({}^{7}\) L. Neuwirth. Proc. Am. Math. Soc., 12, 904 (1961).