Journal of Knot Theory and Its Ramifications

In this paper, we study doubly transitive property of a connected quandle of a prime order. In 2011, Ferman, Nowik and Teicher proved that the group of all quandle automorphisms of a connected quandle of a prime order acts sharply doubly transitively on itself. In this paper, we give a simple alternative proof.

A short, elementary proof is given of the result that the number of components of a link arising from a medial graph M(Γ) by resolving vertices is equal to the nullity of the mod-2 Laplacian matrix of Γ.

We show that the volumes of certain hyperbolic A-adequate links can be bounded (above and) below in terms of two diagrammatic quantities: the twist number and the number of certain alternating tangles in an A-adequate diagram. We then restrict our attention to plat closures of certain braids, a rich family of links whose volumes can be bounded in terms of the twist number alone. Furthermore, in the absence of special tangles, our volume bounds can be expressed in terms of a single stable coefficient of the colored Jones polynomial. Consequently, we are able to provide a new collection of links that satisfy a Coarse Volume Conjecture.

Let L be a quasi-alternating link at a crossing c. We construct an infinite family of quasi-alternating links from L by replacing the crossing c by a product of rational tangles, each of which extends c. Consequently, we determine an infinite family of quasi-alternating Montesinos links. This family includes all classes of quasi-alternating Montesinos links that have been detected by Widmer [Quasi-alternating Montesinos links, J. Knot Theory Ramifications18(10) (2009) 1459–1469]. We conjecture that this family contains all non-alternating quasi-alternating Montesinos links.

We show that a knot has a non-left-orderable surgery if the knot group admits a generalized Baumslag–Solitar relator and satisfies certain conditions on a longitude of the knot. As an application, it is shown that certain positively twisted torus knots admit non-left-orderable surgeries.

A 3-tangle T is the disjoint union of three properly embedded arcs in the unit 3-ball; it is called rational if there is a homeomorphism of pairs from (B³, T) to (D² × I, {x₁, x₂, x₃} × I). Two rational 3-tangles T and T′ are isotopic if there is an orientation-preserving self-homeomorphism h : (B³, T) → (B³, T′) that is the identity map on the boundary. In this paper, we give an algorithm to check whether or not two rational 3-tangles are isotopic by using a modified version of Dehn's method for classifying simple closed curves on surfaces.