Shake slice and shake concordant links

We can construct a [Formula: see text]-manifold by attaching [Formula: see text]-handles to a [Formula: see text]-ball with framing [Formula: see text] along the components of a link in the boundary of the [Formula: see text]-ball. We define a link as [Formula: see text]-shake slice if there exists embedded spheres that represent the generators of the second homology of the [Formula: see text]-manifold. This naturally extends [Formula: see text]-shake slice, a generalization of slice that has previously only been studied for knots, to links of more than one component. We also define a relative notion of shake[Formula: see text]-concordance for links and versions with stricter conditions on the embedded spheres that we call strongly[Formula: see text]-shake slice and strongly[Formula: see text]-shake concordance. We provide infinite families of links that distinguish concordance, shake concordance, and strong shake concordance. Moreover, for [Formula: see text] we completely characterize shake slice and shake concordant links in terms of concordance and string link infection. This characterization allows us to prove that the first non-vanishing Milnor [Formula: see text] invariants are invariants of shake concordance. We also argue that shake concordance does not imply link homotopy.

Tree invariants and Milnor linking numbers with indeterminacy

This paper concerns the tree invariants of string links, introduced by Kravchenko and Polyak, which are closely related to the classical Milnor linking numbers also known as [Formula: see text]-invariants. We prove that, analogously as for [Formula: see text]-invariants, certain residue classes of tree invariants yield link-homotopy invariants of closed links. The proof is arrow diagramatic and provides a more geometric insight into the indeterminacy through certain tree stacking operations. Further, we show that the indeterminacy of tree invariants is consistent with the original Milnor’s indeterminacy. For practical purposes, we also provide a recursive procedure for computing arrow polynomials of tree invariants.

Self C2-equivalence of two-component links and invariants of link maps

We use Kirk’s invariant of link maps [Formula: see text] and its variations due to Koschorke and Kirk–Livingston to deduce results about classical links. Namely, we give a new proof of the Nakanishi–Ohyama classification of two-component links in [Formula: see text] up to [Formula: see text]-link homotopy. We also prove its version for string links, which is due (in a slightly different form) to Fleming–Yasuhara. The proofs do not use Clasper Theory.

Quasi-trivial quandles and biquandles, cocycle enhancements and link-homotopy of pretzel links

We investigate some algebraic structures called quasi-trivial quandles and we use them to study link-homotopy of pretzel links. Precisely, a necessary and sufficient condition for a pretzel link with at least two components being trivial under link-homotopy is given. We also generalize the quasi-trivial quandle idea to the case of biquandles and consider enhancement of the quasi-trivial biquandle cocycle counting invariant by quasi-trivial biquandle cocycles, obtaining invariants of link-homotopy type of links analogous to the quasi-trivial quandle cocycle invariants in Inoue’s paper [A. Inoue, Quasi-triviality of quandles for link-homotopy, J. Knot Theory Ramifications 22(6) (2013) 1350026, doi:10.1142/S0218216513500260, MR3070837].

Detecting Whitney disks for link maps in the four-sphere

It is an open problem whether Kirk’s [Formula: see text]-invariant is the complete obstruction to a link map [Formula: see text] being link homotopic to the trivial link. The link homotopy invariant associates to such a link map [Formula: see text] a pair [Formula: see text], and we write [Formula: see text]. With the objective of constructing counterexamples, Li proposed a link homotopy invariant [Formula: see text] such that [Formula: see text] is defined on the kernel of [Formula: see text] and which also obstructs link null-homotopy. We show that, when defined, the invariant [Formula: see text] is determined by [Formula: see text], and is strictly weaker. In particular, this implies that if a link map [Formula: see text] has [Formula: see text], then after a link homotopy the self-intersections of [Formula: see text] may be equipped with framed, immersed Whitney disks in [Formula: see text] whose interiors are disjoint from [Formula: see text].

A homotopy+ solution to the A-B slice problem

The A-B slice problem, a reformulation of the four-dimensional topological surgery conjecture for free groups, is shown to admit a link-homotopy[Formula: see text] solution. The proof relies on geometric applications of the group-theoretic [Formula: see text]-Engel relation. Implications for the surgery conjecture are discussed.

The universal sl2 invariant and Milnor invariants

The universal [Formula: see text] invariant of string links has a universality property for the colored Jones polynomial of links, and takes values in the [Formula: see text]-adic completed tensor powers of the quantized enveloping algebra of [Formula: see text]. In this paper, we exhibit explicit relationships between the universal [Formula: see text] invariant and Milnor invariants, which are classical invariants generalizing the linking number, providing some new topological insight into quantum invariants. More precisely, we define a reduction of the universal [Formula: see text] invariant, and show how it is captured by Milnor concordance invariants. We also show how a stronger reduction corresponds to Milnor link-homotopy invariants. As a byproduct, we give explicit criterions for invariance under concordance and link-homotopy of the universal [Formula: see text] invariant, and in particular for sliceness. Our results also provide partial constructions for the still-unknown weight system of the universal [Formula: see text] invariant.

On invariants of link maps in dimension four

We fill a gap in the proof that the proposed link homotopy invariant [Formula: see text] of Li is well defined. It is also shown that if the homotopy invariant [Formula: see text] of Schneiderman–Teichner is to be adapted to a link homotopy invariant of link maps, the result coincides with [Formula: see text].