Unexpected connections between Burnside groups and knot theory

Abstract

In classical knot theory and the theory of quantum invariants substantial effort was directed toward the search for unknotting moves on links. We solve, in this article, several classical problems concerning unknotting moves. Our approach uses a concept, Burnside groups of links, that establishes an unexpected relationship between knot theory and group theory. Our method has the potential to be used in computational biology in the analysis of DNA via tangle embedding theory, as developed by D. W. Sumners [Sumners, D. W., ed. (1992) New Scientific Applications of Geometry and Topology (Am Math. Soc., Washington, DC) and Ernst, C. & Sumners, D. W. (1999) Math. Proc. Cambridge Philos. Soc. 126, 23–36].

Connections between knot theory and group theory can be traced back to Listing's pioneering paper of 1847 (1), in which he considered knots and groups of signed permutations. The first well established instance of such a connection was provided by M. Dehn (2). He applied the Poincaré's fundamental group of a knot exterior to study knots and their symmetries. The connection we describe in this article is, on the one hand, deeply rooted in Poincaré's tradition, and, on the other hand, it is unexpected. It was discovered in our study of the cubic skein modules of the three-sphere and led us to the solution of the 20-year-old Montesinos–Nakanishi conjecture.

We outline our main ideas and proofs. The complete exposition of this theory and its applications will be the subject of a future article.

Open Problems

Every link can be simplified to a trivial link by crossing changes (Fig. 1). This observation led to many significant developments, in particular, the construction of the Jones polynomial of links and the Reshetikhin–Turaev invariants of three-manifolds. These invariants had a great impact on modern knot theory. The natural generalization of a crossing change, which is addressed in this article, is a tangle replacement move, that is, a local modification of a link, L, in which a tangle T₁ is replaced by a tangle T₂. Several questions have been asked about which families of tangle moves are unknotting operations.

Fig. 1.

Crossing changes.

One such family of moves, which is significant not only in knot theory but also in computational biology, is the family of rational moves (3). In this paper we devote our attention to special classical cases (20):

1. Nakanishi four-move conjecture, 1979. Every knot is four-move equivalent to the trivial knot.

2. Montesinos–Nakanishi three-move conjecture, 1981. Every link is three-move equivalent to a trivial link.

3. Kawauchi's question, 1985. Are link-homotopic links four-move equivalent?

4. Harikae–Nakanishi conjecture, 1992. Every link is (two, two)-move equivalent to a trivial link.

5. (Two, three)-move question, 1995. Is every link (two, three)-move equivalent to a trivial link?

   The method of Burnside groups, which we introduce, allows us to answer questions ii–v, although conjecture i remains open. We generalize questions ii, iv, and v to the following question:

6. Rational moves question. Is it possible to reduce every link to a trivial link by rational -moves where p is a fixed prime and q is an arbitrary nonzero integer? See Definition 1.1.

To approach our problems, we define invariants of links and call them the Burnside groups of links. These invariants are shown to be unchanged by -rational moves. The strength of our method lies in the fact that we are able to use the well developed theory of classical Burnside groups and their associated Lie rings (4). We first describe, in more detail, how our method is applied to rational moves. In particular, we settle the Montesinos–Nakanishi and Harikae–Nakanishi conjectures. Later, we answer Kawauchi's question in detail.

Definition 1.1: A rational -move¶refers to changing a link by replacing an identity tangle in it by a rational -tangle of Conway (Fig. 2a).

Fig. 2.

Representation of Definition 1.1. (a) Rational move. (b and c) Odd (b) and even (c) rational tangles.

The tangles shown in Fig. 2b are called rational tangles and denoted by T(a₁, a₂,...,a_n) in Conway's notation. A rational tangle is the -tangle if . Conway proved that two rational tangles are ambient isotopic (with boundary fixed) if and only if their slopes are equal (compare ref. 6).

Rational tangles can also be viewed as tangles, which are obtained by applying a finite number of consecutive twists of the neighboring endpoints to the elementary tangle (zero tangle).

Rational -tangles were used by Sumners and Ernst (7) in their mathematical model of DNA recombination. This is a very promising development in computational biology.

The following definition establishes a connection between two classical theories, knot theory and the theory of Burnside groups.∥ Burnside groups of a link play a crucial role in our research and they can contribute significantly to the applications of knot theory in computational biology.

Definition 1.2: Let D be a diagram of a link L. We define the associated core group of D by the following presentation: generators of correspond to arcs of the diagram. Any crossing v_s yields the relation , where y_i corresponds to the overcrossing and y_j, y_k correspond to the undercrossings at v_s (see Fig. 3).

Fig. 3.

Wirtinger's relation.

Remark 1: In the above presentation of one relation may be dropped since it is a consequence of others.

Definition 1.3: (i) The nth Burnside group of a link is the quotient of the fundamental group of the double-branched cover of S³ with the link as the branch set by its subgroup that is generated by all relations of the form wⁿ = 1. Succinctly: .

(ii) The unreduced nth Burnside group of the unoriented link L is the quotient group , where is the associated core group of L.

The relation to the fundamental group of a double-branched cover, mentioned above, is formulated below. An elementary proof using only Wirtinger presentation and also valid for tangles is presented in ref. 12.

Theorem 1.1 (Wada). If D is a diagram of a link (or a tangle) L, then . Furthermore, if we put y_i = 1 for any fixed generator, then reduces to .

The next theorem allows us to use Burnside groups to analyze elementary moves on links.

Theorem 1.2. The groups B_L(n) and B̂_L(n) are preserved by rational -moves. In particular, the nth Burnside group is preserved by n-moves.

Proof: Let L′ be obtained from L by a -move. Then is obtained from by performing the -surgery. Such a surgery can be easily proved to preserve the nth Burnside group of the fundamental group of the manifold.

Reductions by Rational Moves

We show that the answer to the rational move question, problem (vi), is negative.

Theorem 2. (i) The closure, , of the three-braid (Fig. 4a) is not -move reducible to a trivial link for any prime number p ≥ 5.

Fig. 4.

Braids representing two full twists. (a) Three-braid. (b) Five-braid.

(ii) The closure of the five-braid (Fig. 4b) is not three-move reducible to a trivial link.

Sketch of the Proof: We use Sanov's theorem about the structure of the Lie algebra associated to the Burnside group of prime exponent p ≥ 5 (13, 14). For p = 3 we observe that the third Burnside group is the quotient of the free Burnside group, B(4, 3), by the normal subgroup generated by relations , where the words Q_i can be computed from Fig. 4b by using the core relations (Fig. 3).†† To see that relations are nontrivial in the free Burnside group, we use the theorem of Levi and van der Waerden about the structure of the associated Lie rings of Burnside groups of exponent 3 (16).

The closed braid is three-move equivalent to a link of 20 crossings [closure of the five-string braid ]. It is still an open problem whether the Montesinos–Nakanishi conjecture holds for links up to 19 crossings. Q. Chen proved that it holds for links up to 12 crossings (17).

The negative answers to problems ii, iii, and v follow from the theorem for p = 3, 5, and 7, respectively (see ref. 18 for details).

Kawauchi's Question on Four-Moves

Here, we use the fourth Burnside group of links to show that there is an obstruction to four-move reducibility of links that are link homotopically trivial. Therefore, the answer to Kawauchi's question is negative.

Let w denote the “half” two-cabling of the Whitehead link, which is the link homotopy equivalent to the trivial link of three components (Fig. 5a).

Fig. 5.

Half two-cabling of the Whitehead link and its two-tangle presentation. (a) The diagram of w.(b) Computation of relations of B_w(4).

Theorem 3. The link w is not four-move equivalent to a trivial link.

Our proof uses the obstruction in the Burnside group B(2, 4). The obstruction lies in the last nontrivial term of the lower central series, γ₅, of B(2, 4) or equivalently in the fifth term of the associated graded Lie ring of B(2, 4). In fact, we have:

Lemma. B_w(4) = B(2, 4)/γ₅.

Sketch of the Proof: The lower central series of B(2, 4) is known to be of class 5, with the last term γ₅ = L₅ isomorphic to Z₂ ⊕ Z₂ (14). First, we compute by using a presentation of the associated core group of w by putting the generator z = 1. From the diagram we obtain the relations: Q₁x^–1 and Q₂y^–1 (Fig. 5b), and further the equivalent relations

Therefore

In B_w(4) this relations can be reduced into:

Analogously, R₂ = [y, x, y, x, y] ∈ γ₅.

Those elements form a basis of γ₅ = Z₂ ⊕ Z₂ considered as a Z₂ linear space. We verified this fact by using the programs gap, magnus, and magma. These calculations can be done manually; however, they strongly depend on the unpublished Ph.D. thesis of J. J. Tobin (19), as it was pointed out to us by M. Vaughan-Lee (University of Oxford, Oxford).

It follows from the above Lemma, and the fact that γ₅ is in the center of B(2, 4), that |B_w(4)| = 2¹⁰. On the other hand, the abelianization of B_w(4) is isomorphic to Z₄ ⊕ Z₄. So if w were four-reducible into a trivial link then the trivial link would have three components. But B_T3(4) = B(2, 4) has 2¹² elements, as predicted by Burnside (8) and verified by Tobin (19). This completes the proof.

Remark 2: Nakanishi showed, using Alexander modules, that the Borromean rings, BR, cannot be reduced to a trivial link by four-moves. Our Burnside obstruction method also works in this case. Knowing that |B(2, 4)| = 2¹² we can verify that |B_BR(4)| = 2⁵. In addition, we conclude that w and BR are not four-move equivalent.

By Coxeter's theorem (21) the quotient group is finite. Therefore, for closed three-braids, we can list all possible fourth Burnside groups. This allows us to find all four-move equivalence classes of closed three-braids (unpublished work).

Limitations of the Burnside Group Invariant

The method based on the Burnside group invariant has been quite successful in the study of the unknotting property of several classes of tangle replacement moves. As we saw earlier, for any fixed prime number p ≥ 3 and an arbitrary nonzero integer q, rational -moves are not unknotting operations. However, our method has its limitations. We have been unable to find, by our method, obstructions for -reduction of a link L to a trivial link if the abelianization of the nth Burnside group of the link, , is a cyclic group (i.e., {1} or Z_n). This is explained in Theorem 4.

Define the restricted Burnside group of a link, R_L(n), as the quotient group B_L(n)/N, where N is the intersection of all normal subgroups of B_L(n) of finite indexes.

Theorem 4. Let n be a power of a prime number. Assume that the abelianization of B_L(n) is a cyclic group. Then the restricted Burnside group, R_L(n), is isomorphic to . In particular, if B_L(n) is finite (e.g., for n = 2, 3, 4), then B_L(n) is a cyclic group.

Proof: It was proved by E. Zelmanov (14) that R(r, n) is finite for any n. It follows that R_L(n) is finite for any n.

Let γ₁ ≥ γ₂ ≥ ··· ≥ γ_i ≥ ··· be the lower central series of B_L(n). If (B_L(n))^(ab) [and therefore also (R_L(n))^(ab)] is a cyclic group then γ₂ = γ₃ = γ₄.... Since R_L(n) is a finite nilpotent group, therefore R_L(n) = (B_L(n))^(ab).

If B_L(n) is finite (as it is in the case of n = 2, 3, 4), then R_L(n) = B_L(n), so B_L(n) is a cyclic group.

Therefore, the method based on the Burnside group invariant will not produce any obstructions for the Nakanishi four-move conjecture and the Kawauchi four-move question for a link of two components.

Conclusion

Our interest in the analysis of rational moves on links was inspired by our long pursuit of a program for understanding a 3D manifold by the knot theory that it supports. The method we introduced, the Burnside group of links, not only settles classical conjectures (e.g., the Montesinos–Nakanishi conjecture) but also has clear potential to be used in computational biology in an analysis of DNA, its recombination, action of topoisomers, and analysis of protein folding and protein evolution.