Satellite knots and trivializing bands

We show an infinite family of satellite knots that can be unknotted by a single band move, but such that there is no band unknotting the knots which is disjoint from the satellite torus.

The Neuwirth Conjecture for a family of satellite knots

Let [Formula: see text] be a nontrivial knot in [Formula: see text]. It was conjectured that there exists a Neuwirth surface for [Formula: see text]. That is, a closed surface in [Formula: see text] containing the knot [Formula: see text] as a nonseparating curve and such that every compressing disk for the surface intersects the knot in at least two points. We provide explicit constructions of Neuwirth surfaces for a family of satellite knots, which do not depend on the existence of nonorientable algebraically incompressible and [Formula: see text]-incompressible spanning surfaces for these knots.

On lassos and the Jones polynomial of satellite knots

In this paper, I present a new family of knots in the solid torus called lassos, and their properties. Given a knot [Formula: see text] with Alexander polynomial [Formula: see text], I then use these lassos as patterns to construct families of satellite knots that have Alexander polynomial [Formula: see text] where [Formula: see text]. In particular, I prove that if [Formula: see text] these satellite knots have different Jones polynomials.

ON THE KNOT FLOER HOMOLOGY OF A CLASS OF SATELLITE KNOTS

Knot Floer homology is an invariant for knots in the three-sphere for which the Euler characteristic is the Alexander–Conway polynomial of the knot. The aim of this paper is to study this homology for a class of satellite knots, so as to see how a certain relation between the Alexander–Conway polynomials of the satellite, companion and pattern is generalized on the level of the knot Floer homology. We also use our observations to study a classical geometric invariant, the Seifert genus, of our satellite knots.