Groups of the virtual trefoil and Kishino knots

Abstract

In the paper [13] for an arbitrary virtual link L three groups G 1,r (L), r > 0, G 2 (L) and G 3 (L) were defined. In the present paper these groups for the virtual trefoil are investigated. The structure of these groups are found out and the fact that some of them are not isomorphic to each other is proved. Also we prove that G 3 distinguishes the Kishino knot from the trivial knot. The fact that these groups have the lower central series which does not stabilize on the second term is noted. Hence we have a possibility to study these groups using quotients by terms of the lower central series and to construct representations of these groups in rings of formal power series. It allows to construct an invariants for virtual knots.In the present work the fact that groups of the virtual trefoil T v G 1,r (T v ) and G 2 (T v ) are not isomorphic to each other will be proven.Note that the most of virtual knot groups do not distinguish the group of the Kishino knot from the trivial one. The group defined in [9] is an exception. In the paper [13] the fact that groups constructed using representations W 1,r and W 2 do not distinguish the Kishino knot from the trivial one is stated. And the natural question whether the group constructed using the representation W 3 distinguishes the Kishino knot from the trivial one was formulated in [13].In present work the positive answer on the question is given.As it is well known if G is a group of a classical knot, then its commutator subgroup coincides with the third term of lower central series. Therefore, the factorization by terms of the lower central series cannot be used for distinguish of knots. On the other hand as it was pointed out in the paper [3] some of virtual link groups are residually nilpotent groups. Recall that a group G is referred to as residually nilpotent if for every nontrivial element g from G there exists a homomorphism ϕ : G −→ N on a nilpotent group N such that ϕ(g) = 1. It is easy to observe that G is residually nilpotent if and only if the intersection of all terms of its lower central series is a trivial group.In the paper [6] was noted that the virtual trefoil group from [1] has different first five terms of lower central series. This fact allows to construct new invariants of virtual links. The question occurs whether virtual link groups are residually nilpotent or not. Moreover in [6] representations of some virtual knot groups into finite algebras were constructed. In the present paper this approach is being developed.Let us describe the paper content. In Section 2 we recall representations constructed in [13] of the virtual braid group V B n into the group of automorphisms of free groups which are extensions of the Wada representations of braid group B n . Also we give presentations of three groups G 1,r , G 2 and G 3 of the virtual trefoil. In Section 3 the structure of these groups is found out. In Section 4, using quotients by the second commutator subgroups, we prove that G 1,r is not isomorphic to G 2 ; we study the quotients by the terms of t...