Yasutaka Nakanishi asked in 1981 whether a 3-move is an unknotting operation. In Kirby's problem list, this question is called The Montesinos-Nakanishi 3-move conjecture. We define the nth Burnside group of a link and use the 3rd Burnside group to answer Nakanishi's question; ie, we show that some links cannot be reduced to trivial links by 3-moves.

Abstract. We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched coverings of S 3 branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable. Many other examples of non-orderable groups are obtained by taking 3-fold branched covers of S 3 branched along various hyperbolic 2-bridge knots. The manifold obtained in such a way from the 5 2 knot is of special interest as it is conjectured to be the hyperbolic 3-manifold with the smallest volume.

The existence of periodic solutions in Γ-symmetric Newtonian systemsẍ = −∇f (x) can be effectively studied by means of the Γ × O(2)-equivariant gradient degree with values in the Euler ring U (Γ × O(2)). In this paper we show that in the case of Γ being a finite group, the Euler ring U (Γ × O(2)) and the related basic degrees are effectively computable using Euler ring homomorphisms, the Burnside ring A(Γ × O(2)), and the reduced Γ × O(2)-degree with no free parameters. We present several examples of Newtonian systems with various symmetries, for which we show existence of multiple periodic solutions. We also provide exact value of the equivariant topological invariant for those problems.1 for example a difference of the equivariant gradient degrees on large and small ball). Such cases are for example when the system (1) is asymptotically linear or satisfy a Nagumo-type growth condition. Then clearly the existence of non-trivial solutions (i.e. outside the small ball) can be concluded by the fact that ω = 0. However, one can be also interested to predict the existence of multiple 2π-periodic solutions with different types of symmetry. In such a case, the coefficients of ω corresponding to the so-called maximal orbit type can provide the crucial information in order to formulate such results. But the maximality of such obit types implies that it is a generator of the Burnside ring A(G), therefore it can actually be detected by the Γ × O(2)-equivariant degree with no free parameter, which can be much easier computed than the equivariant gradient degree. Similar arguments apply to the system (2), which we can consider as a bifurcation problem with a parameter λ. More precisely, in this case we are looking for critical values λ o of the parameter λ > 0, to which we can associate the Γ × O(2)-equivariant gradient bifurcation invariants ω(λ o ) ∈ U (Γ×O(2)) classifying the bifurcation of 2π-periodic solutions from the zero solution. The existence and multiplicity of such bifurcating branches of 2π-periodic solutions can be described from the information contained in the invariants ω(λ o ). Consequently, all the essential information needed to establish the existence and multiplicity results for the systems (1) and (2) can be extracted from the Γ × O(2)-equivariant degree (with no free parameter) of J which takes values in the Burnside ring A(G). It is clear that the Γ × O(2)-equivariant degree without free parameter can be easily computed (without getting entangled in complicated technical details), has similar properties and provides enough information for analyzing these problems.Nevertheless, let us emphasize that only the equivariant invariants ω ∈ U (G) (without truncation of its coefficients) provide a complete equivariant topological classification for the related solution sets to (1) or (2).To illustrate the usage and the computations of the associated with the systems (1) and (2) equivariant invariants, in section 7 we present several examples of symmetric Newtonian systems, for which the exact values of the a...

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 C onnections 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 ProblemsEvery 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 T 1 is replaced by a tangle T 2 . Several questions have been asked about which families of tangle moves are unknotting operations.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) 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 p q -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 MontesinosNakanishi and Harikae-Nakanishi conjectures. Later, we answer Kawauchi's question in detail. Definition 1.1: A rational p q -move ¶ refers to changing a link by replacing an identity tangle in it by a rational p q -tangle of Conway (Fig. 2a).The tangles shown in Fig. 2b are called rational tangles and denoted by T(a 1 , a 2 , . . . , a n ) in Conway's notation. A rational tangle is the p q -tangle if p q ϭ a n ϩ 1͞(a nϪ1 ϩ ⅐ ⅐ ⅐ ϩ (1͞a 1 )). Conway proved that two rational tangles are ambient isotopic (with boundary fix...

A left order on a magma (e.g., semigroup) is a total order of its elements that is left invariant under the magma operation. A natural topology can be introduced on the set of all left orders of an arbitrary magma. We prove that this topological space is compact. Interesting examples of nonassociative magmas, whose spaces of right orders we analyze, come from knot theory and are called quandles. Our main result establishes an interesting connection between topological properties of the space of left orders on a group, and the classical algebraic result by Conrad [4] and Loś [13] concerning the existence of left orders.