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 2-GENERALIZED Knot Group Determines the Knot
Communications in Contemporary Mathematics, 2008
Generalized knot groups Gn(K) were introduced independently by Kelly [5] and Wada [10]. We prove that G2(K) determines the unoriented knot type and sketch a proof of the same for Gn(K) for n > 2.
2018
We prove that given a Conway algebraic link diagram D with n crossings then D can be embedded on the cubic lattice with a length bounded above by en, where cis a positive constant independent of D and n. This implies that the ropelength of alternating Conway algebraic knots growths at most linear with their crossing number.
Knots are determined by their complements
Bulletin of The American Mathematical Society, 1989
Two (smooth or PL) knots K, K' in S 3 are equivalent if there exists a homeomorphism h: S 3 -• S 3 such that h(K) = K'. This implies that their complements S 3 -K and S 3 -K' are homeomorphic. Here we announce the converse implication. THEOREM 1. If two knots have homeomorphic complements then they are equivalent. This answers a question apparently first raised by Tietze [T, p. 83]. It was previously known that there were at most two knots with a given complement [CGLS, Corollary 3]. Whitten [W] has shown that prime knots with isomorphic groups have homeomorphic complements. Hence we have COROLLARY 1.1. If two prime knots have isomorphic groups then they are equivalent.