The 2-generalized Knot Group Determines the Knot

Generalized knot groups Gn(K) were introduced independently by Kelly (1991) and Wada (1992). We prove that G2(K) determines the unoriented knot type and sketch a proof of the same for Gn(K) for n > 2. 1. The 2–generalized knot group Generalized knot groups were introduced independently by Kelly [5] and Wada [10]. Wada arrived at these group invariants of knots by searching for homomorphisms of the braid group Bn into Aut(Fn), while Kelley’s work was related to knot racks or quandles [1, 4] and Wirtinger-type presentations. The Wirtinger presentation of a knot group expresses the group by generators x1, . . . , xk and relators r1, . . . , rk−1, in which each ri has the form x j xix ∓1 j x −1 i+1 for some permutation i 7→ j of {1, . . . , k} and map {1, . . . , k} → {±1}. The group Gn(K) is defined by replacing each ri by x j xix ∓n j x −1 i+1 . In particular, G1(K) is the usual knot group. In [9], responding to a preprint of Xiao-Song Lin and the first author [6], Tuffley showed that Gn(K) distinguishes the square and granny knots. Gn(K) cannot distinguish a knot from its mirror image. But G2(K) is, in fact, a complete unoriented knot invariant. Theorem 1.1. The 2–generalized knot group G2(K) determines the knot K up to reflection. We will assume K is a non-trivial knot in the following proof, although it is not essential. It is clear from the proof that the trivial knot is the only knot with G2(K) = Z. Wada described Gn(K) as the fundamental group of the space Mn(K) obtained by gluing the boundary torus of the knot exterior to another torus by the map S×S → S×S defined by f(z1, z2) = (z n 1 , z2), where z1 represents the meridian and z2 represents a longitude. We will use this description. We will call the glued-on torus the core torus. Note that M2(K) is a closed manifold: it can be described as the result of gluing Mb × S into the knot exterior, where Mb denotes the Möbius band. It is clearly Haken, since its fundamental group has a Z quotient, and it is irreducible and P–irreducible since its orientation cover is the double of the knot exterior