STRONGLY n-TRIVIAL KNOTS (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.
Proceedings of the American Mathematical Society, 1987
The class of knots consisting of twisted Whitehead doubles can have arbitrarily large free genus but all have genus 1.
Journal of Differential Geometry
This paper studies knots that are transversal to the standard contact structure in R 3 , bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type T K is transversally simple if it is determined by its topological knot type K and its Bennequin number. The main theorem asserts that any T K whose associated K satisfies a condition that we call exchange reducibility is transversally simple. As a first application, we prove that the unlink is transversally simple, extending the main theorem in [10]. As a second application we use a new theorem of Menasco [17] to extend a result of Etnyre [11] to prove that all iterated torus knots are transversally simple. We also give a formula for their maximum Bennequin number. We show that the concept of exchange reducibility is the simplest of the constraints that one can place on K in order to prove that any associated T K is transversally simple. We also give examples of pairs of transversal knots that we conjecture are not transversally simple.
The Unknotting Number of Some Knots
International journal of pure and applied mathematics, 2015
We compute the unknotting number for infinite families of knots by using a famous inequality due to Murasugi that relates the unknotting number of a knot to the signature of the same knot. Also, we determine the unknotting number and show it is equal to two for some knots in the knot table with twelve crossings or less by another inequality due to Nakanishi that relates the unknotting number of a knot to the surgerical description number of the knot and by a theorem that is due to Kanenobu and Murakami.
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.
Knots of Ten or Fewer Crossings of Algebraic Order Two
2012
A knot K is an S1 embedded in S3. If K is the boundary of a D2 properly embedded in D4, we call that knot slice. The set of knots modulo slice knots is called the knot concordance group: this is a group under connected sums, with the orientation–reversed mirror of a knot being its inverse. Levine defined a group called the algebraic concordance group of Witt classes of Seifert matrices of knots, and proved that this group is isomorphic to Z ∞ ⊕ Z ∞ 2 ⊕Z∞4. He also showed that there is a surjective homomorphism from the knot concordance group to the algebraic concordance group [5]. Casson and Gordon proved that the kernel of this map was non-trivial [1]. In this paper we investigate knots in the knot concordance group that represent elements of order two in the algebraic concordance group. We prove that many are not of knot concordance order two. The orders of many knots in the algebraic concordance group have been calculated. Morita [9] and Kawauchi [3] have published tables of all ...