Journal of Knot Theory and Its Ramifications

As an example of the transitions between some of the eight geometries of Thurston, investigated in [M. T. Lozano and J. M. Montesinos-Amilibia, On the degeneration of some 3-manifold geometries via unit groups of quaternion algebras RACSAM 109 (2015) 669–715], we study the geometries supported by the conemanifolds obtained by surgery on the trefoil knot with singular set the core of the surgery. The geometric structures are explicitly constructed. The most interesting phenomenon is the transition from SL(2, ℝ)-geometry to S³-geometry through Nil-geometry. A plot of the different geometries is given, in the spirit of the analogous plot of Thurston for the geometries supported by surgeries on the figure-eight knot.

We show that the category of quandles satisfies a Maltsev property relative to a certain class of split epimorphisms. This structural aspect produces a class of congruences, called puncturing, which have the property to permute with any other congruence.

We consider a braid β which acts on a punctured plane. Then we construct a local system on this plane and find a homology cycle D in its symmetric power, such that D ⋅ β(D) coincides with the Alexander polynomial of the plait closure of β.

We consider the group of unrestricted virtual braids, describe its structure and explore its relations with fused links. Also, we define the groups of flat virtual braids and virtual Gauss braids and study some of their properties, in particular their linearity.

We classify Legendrian torus knots in S¹ × S² with its standard tight contact structure up to Legendrian isotopy.

It is known that the Burau representation of the 4-strand braid group is faithful if and only if certain matrices f and k generate a (non-abelian) free group. Regarding f and k as isometries of a Euclidean building, we show that f³ and k³ generate a free group. We give two proofs, one utilizing the metric geometry of the building, and the other using simplicial retractions.