Narrow Results

We study the existence of relations between the degrees of the knot polynomials and some classical knot invariants, partially confirming and extending the question of Morton on the skein polynomial and a recent question of Ferrand.

We prove a simple necessary and sufficient condition for a two-bridge knot $K(p,q)$ to be quasipositive, based on the continued fraction expansion of $p/q$. As an application, coupled with some classification results in contact and symplectic topology, we give a new proof of the fact that smoothly slice two-bridge knots are non-quasipositive. Another proof of this fact using methods within the scope of knot theory is presented in Appendix A, by Stepan Orevkov.

We use the Bar-Natan Ж-correspondence to identify the generalized Alexander polynomial of a virtual knot with the Alexander polynomial of a two component welded link. We show that the Ж-map is functorial under concordance, and also that Satoh’s Tube map (from welded links to ribbon knotted tori in $S^4$) is functorial under concordance. In addition, we extend classical results of Chen, Milnor and Hillman on the lower central series of link groups to links in thickened surfaces. Our main result is that the generalized Alexander polynomial vanishes on any knot in a thickened surface which is virtually concordant to a homologically trivial knot. In particular, this shows that it vanishes on virtually slice knots. We apply it to complete the calculation of the slice genus for virtual knots with four crossings and to determine non-sliceness for a number of 5-crossing and 6-crossing virtual knots.

Knot concordance plays a crucial role in the low-dimensional topology. We propose a very elementary techniques which allows one to construct a lot of sliceness obstructions for knots in the full torus. Our approach deals with group theoretical techniques; it is completely combinatorial, and the groups are very easy to deal with.

Contrary to the case of knots, the property that a link has an unlinking number one is reflected in its Conway polynomial. Examples of the determination of the unlinking number are given.