The 2-GENERALIZED Knot Group Determines the Knot (original) (raw)
Journal of Knot Theory and Its Ramifications, 2008
Generalized knot groups Gn(K) were introduced first by Wada and Kelly independently. The classical knot group is the first one G1(K) in this series of finitely presented groups. For each natural number n, G1(K) is a subgroup of Gn(K) so the generalized knot groups can be thought of as extensions of the classical knot group. For the square knot SK and the granny knot GK, we have an isomorphism G1(SK) ≅ G1(GK). From the presentations of Gn(SK) and Gn(GK), for n > 1, it seems unlikely that Gn(SK) and Gn(GK) would be isomorphic to each other. Curiously, we are able to show that for any finite group H, the numbers of homomorphisms from Gn(SK) and Gn(GK) to H, respectively, are the same. Moreover, the numbers of conjugacy classes of homomorphisms from Gn(SK) and Gn(GK) to H, respectively, are also the same. It remains a challenge to us to show, as we would like to conjecture, that Gn(SK) and Gn(GK) are not isomorphic to each other for all n > 1.
arXiv (Cornell University), 2021
For a knot diagram K, the classical knot group π1(K) is a free group modulo relations determined by Wirtinger-type relations on the classical crossings. The classical knot group is invariant under the Reidemeister moves. In this paper, we define a set of quotient groups associated to a knot diagram K. These quotient groups are invariant under the Reidemeister moves and the set includes the extended knot groups defined by Boden et al and Silver and Williams.
On the classification of knots
Proceedings of the American Mathematical Society, 1974
Linking numbers between branch curves of irregular covering spaces of knots are used to extend the classification of knots through ten crossings and to show that the only amphicheirals in Reidemeister’s table are the seven identified by Tait in 1884. Diagrams of the 165 prime 10 10 -crossing knot types are appended. (Murasugi and the author have proven them prime; Conway claims proof that the tables are complete.) Including the trivial type, there are precisely 250 prime knots with ten or fewer crossings.
The homotopy groups of knots. I. How to compute the algebraic 2-type
Pacific Journal of Mathematics, 1981
Let K be a CW complex with an aspherical splitting, i.e., with subcomplexes ϋΓ_ and K + such that (a) K-K-UK+ and (b) iΓ_, K 0 =K-ΠK +1 K + are connected and aspherical. The main theorem of this paper gives a practical procedure for computing the homology H*K of the universal cover K of K. It also provides a practical method for computing the algebraic 2-type of K 9 i.e., the triple consisting of the fundamental group π x K, the second homotopy group π 2 K as a TΓilΓ-module, and the first /^-invariant kK. The effectiveness of this procedure is demonstrated by letting K denote the complement of a smooth 2-knot (S 4 , JcS 2). Then the above mentioned methods provide a way for computing the algebraic 2-type of 2-knots, thus solving problem 36 of R. H. Fox in his 1962 paper, "Some problems in knot theory." These methods can also be used to compute the algebraic 2-type of 3-manifolds from their Heegaard splittings. This approach can be applied to many other well known classes of spaces. Various examples of the computation are given.