Orderability of knot quandles

2006

In this paper we define a partial order on the set of all knots and links using a special property derived from their minimal diagrams. A knot or link K 0 is called a predecessor of a knot or link K if Cr(K 0 ) < Cr(K) and a diagram of K 0 can be obtained from a minimal diagram D of K by a single crossing change. In such a case we say that K 0 < K. We investigate the sets of knots that can be obtained by single crossing changes over all minimal diagrams of a given knot. We show that these sets are specific for dierent knots and permit partial ordering of all the knots. Some interesting results are presented and many questions are posed.

We show that any two diagrams of the same knot or link are connected by a sequence of Reidemeister moves which are sorted by type.

Journal of Knot Theory and Its Ramifications, 2015

We show that the fundamental group of the double branched cover of an infinite family of homologically thin, non-quasi-alternating knots is not left-orderable, giving further support for a conjecture of Boyer, Gordon, and Watson that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable.

Algebraic & Geometric Topology, 2005

We define enhanced presentations of quandles via generators and relations with additional information comprising signed operations and an order structure on the set of generators. Such a presentation determines a virtual link diagram up to virtual moves. We list formal Reidemeister moves in which Tietze moves on the presented quandle are accompanied by corresponding changes to the order structure. Omitting the order structure corresponds to replacing virtual isotopy by welded isotopy.

International Journal of Mathematics, 2021

We consider the classical pretzel knots [Formula: see text], where [Formula: see text] are positive odd integers. By using continuous paths of elliptic [Formula: see text]-representations, we show that (i) the 3-manifold obtained by [Formula: see text]-surgery on [Formula: see text] has left orderable fundamental group if [Formula: see text] and (ii) the [Formula: see text]-cyclic branched cover of [Formula: see text] has left orderable fundamental group if [Formula: see text].

Topology, 1985

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.

Osaka Journal of Mathematics, 2008

We prove that the fundamental quandle of the trefoil knot is isomorphic to the projective primitive subquandle of transvections of the sy mplectic space Z Z. The last quandle can be identified with the Dehn quandle of the tor us and the cord quandle on a 2-sphere with four punctures. We also show that the fundamental quandle of the long trefoil knot is isomorphic to the cord quandle on a 2-sphere with a hole and three punctures.

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.

Journal of Knot Theory and Its Ramifications

Let [Formula: see text] be ribbon knottings of [Formula: see text]-spheres or tori in [Formula: see text], [Formula: see text]. We show that if the knot quandles of these knots are isomorphic, then the ribbon knottings are stably equivalent, in the sense of Nakanishi and Nakagawa, after taking a finite number of connected sums with trivially embedded copies of [Formula: see text].

Contemporary mathematics, 2017

2008

In this paper we give combinatorial proofs of the classification of unoriented and oriented rational knots based on the now known classification of alternating knots and the calculus of continued fractions. We also characterize the class of strongly invertible rational links. Rational links are of fundamental importance in the study of DNA recombination. AMS Subject Classification: 57M27

In this paper we give combinatorial proofs of the classification of unoriented and oriented rational knots based on the now known classification of alternating knots and the calculus of continued fractions. We also characterize the class of strongly invertible rational links. Rational links are of fundamental importance in the study of DNA recombination.

Proceedings of the American Mathematical Society, 2019

In this note, residual finiteness of quandles is defined and investigated. It is proved that free quandles and knot quandles of tame knots are residually finite and Hopfian. Residual finiteness of quandles arising from residually finite groups (conjugation, core, and Alexander quandles) is established. Further, residual finiteness of automorphism groups of some residually finite quandles is also discussed.

2009

In this paper we give combinatorial proofs of the classification of unoriented and oriented rational knots based on the now known classification of alternating knots and the calculus of continued fractions. We also characterize the class of strongly invertible rational links. Rational links are of fundamental importance in the study of DNA recombination. AMS Subject Classification: 57M27