Virtual parity Alexander polynomial

In this paper, we define the virtual Alexander polynomial following the works of Boden et al. (2016) [Alexander invariants for virtual knots, J. Knot Theory Ramications 24(3) (2015) 1550009] and Kaestner and Kauffman [Parity biquandles, in Knots in Poland. III. Part 1, Banach Center Publications, Vol. 100 (Polish Academy of Science Mathematical Institute, Warsaw, 2014), pp. 131–151]. The properties of this invariant are explored and some examples are computed. In particular, the invariant demonstrates that many virtual knots cannot be unknotted by crossing changes on only odd crossings.

Embedding spheres in knot traces

The trace of the $n$ -framed surgery on a knot in $S^{3}$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded $2$ -sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable $3$ -dimensional knot invariants. For each $n$ , this provides conditions that imply a knot is topologically $n$ -shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.

Open Problems

Open problems in the study of topological 4-manifolds are explained in detail. An important open problem is to determine whether the disc embedding theorem and its antecedents hold for all groups; in other words, whether all groups are good. The disc embedding conjecture and the surgery conjecture are stated. The relationships between these conjectures and their various reformulations are explained. Of particular interest are the reformulations in terms of freely slicing certain infinite families of links. In particular, the surgery conjecture is true if and only if all good boundary links are freely slice. Good boundary links are the many-component analogues of Alexander polynomial one knots.

Alexander polynomial and spanning trees

Inspired by the combinatorial constructions in earlier work of the authors that generalized the classical Alexander polynomial to a large class of spatial graphs with a balanced weight on edges, we show that the value of the Alexander polynomial evaluated at [Formula: see text] gives the weighted number of the spanning trees of the graph.

Virtual concordance and the generalized Alexander polynomial

We use the Bar-Natan [Formula: see text]-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 [Formula: see text]-map is functorial under concordance, and also that Satoh’s Tube map (from welded links to ribbon knotted tori in [Formula: see text]) 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.

On Fox’s trapezoidal conjecture for closed 3-braids

An old and challenging conjecture proposed by Fox in 1962 states that the coefficients of the Alexander polynomial of any alternating knot are trapezoidal. In other words, these coefficients, increase, stabilize then decrease in a symmetrical way. This curious behavior of the Alexander polynomial has been confirmed in several special cases of alternating knots. The purpose of this paper is to prove that the conjecture holds for some families of alternating knots of braid index 3.