Narrow Results

Given a knot K we may construct a group G_n(K) from the fundamental group of K by adjoining an nth root of the meridian that commutes with the corresponding longitude. These "generalized knot groups" were introduced independently by Wada and Kelly, and contain the fundamental group as a subgroup. The square knot SK and the granny knot GK are a well-known example of a pair of distinct knots with isomorphic fundamental groups. We show that G_n(SK) and G_n(GK) are non-isomorphic for all n ≥ 2. This confirms a conjecture of Lin and Nelson, and shows that the isomorphism type of G_n(K), n ≥ 2, carries more information about K than the isomorphism type of the fundamental group. The appendix contains some results on representations of the trefoil group in PSL(2, p) that are needed for the proof.

It is known that every nontrivial knot has at least two quadrisecants. Given a knot, we mark each intersection point of each of its quadrisecants. Replacing each subarc between two nearby marked points with a straight line segment joining them, we obtain a polygonal closed curve which we will call the quadrisecant approximation of the given knot. We show that for any hexagonal trefoil knot, there are only three quadrisecants, and the resulting quadrisecant approximation has the same knot type.

We go along a knot diagram, and get a sequence of over- and under- crossing points. We will study which kinds of sequences are realized by diagrams of the trefoil knot. As an application, we will characterize the Shimizu warping polynomials for diagrams of the trefoil knot.

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.