Narrow Results

The universal $s{l}_2$ invariant of string links has a universality property for the colored Jones polynomial of links, and takes values in the $\hslash$-adic completed tensor powers of the quantized enveloping algebra of $s{l}_2$. In this paper, we exhibit explicit relationships between the universal $s{l}_2$ 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 $s{l}_2$ 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 $s{l}_2$ invariant, and in particular for sliceness. Our results also provide partial constructions for the still-unknown weight system of the universal $s{l}_2$ invariant.

This paper is a follow-up to [10], in which the author showed that the only real-valued finite type invariants of link homotopy are the linking numbers of the components. In this paper, we extend the methods used to show that the only real-valued finite type invariants of link concordance are, again, the linking numbers of the components.

We study the effect of mutation on link concordance and 3-manifolds. We show that the set of links concordant to sublinks of homology boundary links is not closed under positive mutation. We also show that mutation does not preserve homology cobordism classes of 3-manifolds. A significant consequence is that there exist 3-manifolds which have the same quantum SU(2)-invariants but are not homology cobordant. These results are obtained by investigating the effect of mutation on the Milnor -invariants, or equivalently the Massey products.

We show that the twisted signature invariants of boundary link concordance derived from unitary representations of the free group are actually invariants of ordinary link concordance. We also show how the discontinuity locus of this signature function is determined by Seifert matrices of the link.

We give new examples of 2-component links with linking number one and unknotted components that are topologically concordant to the positive Hopf link, but not smoothly so – in addition, they are not smoothly concordant to the positive Hopf link with a knot tied in the first component. Such examples were previously constructed by Cha–Kim–Ruberman–Strle; we show that our examples are distinct in smooth concordance from theirs.

We establish several results about two short exact sequences involving lower terms of the $n$-solvable filtration, $\{{{ℱ}}_n^m\}$ of the string link concordance group $C^m$. We utilize the Thom–Pontryagin construction to show that the Sato–Levine invariants ${\bar{\mu }}_{(iijj)}$ must vanish for 0.5-solvable links. Using this result, we show that the short exact sequence $0\to {{ℱ}}_0^m/{{ℱ}}_{0.5}^m\to {{ℱ}}_{-0.5}^m/{{ℱ}}_{0.5}^m\to {{ℱ}}_{-0.5}^m/{{ℱ}}_0^m\to 0$ does not split for links of two or more components, in contrast to the fact that it splits for knots. Considering lower terms of the filtration $\{{{ℱ}}_n^m\}$ in the short exact sequence $0\to {{ℱ}}_{-0.5}^m/{{ℱ}}_0^m\to {{𝒞}}^m/{{ℱ}}_0^m\to {{𝒞}}^m/{{ℱ}}_{-0.5}^m\to 0$, we show that while the sequence does not split for $m\ge 3$, it does indeed split for $m=2$. This allows us to determine that the quotient ${{𝒞}}^2/{{ℱ}}_0^2\cong {\mathbb{Z}}_2\oplus {\mathbb{Z}}_2\oplus {\mathbb{Z}}_2\oplus \mathbb{Z}$.

We show that Cochran's invariants ${\beta }^i(L)$ of a $2$-component link $L$ in the $3$-sphere can be computed as intersection invariants of certain 2-complexes in the $4$-ball with boundary $L$. These 2-complexes are special types of twisted Whitney towers, which we call Cochran towers, and which exhibit a new phenomenon: A Cochran tower of order $2k$ allows the computation of the ${\beta }^i$ invariants for all $i\le k$, i.e. simultaneous extraction of invariants from a Whitney tower at multiple orders. This is in contrast with the order $n$ Milnor invariants (requiring order $n$ Whitney towers) and consistent with Cochran's result that the ${\beta }^i(L)$ are integer lifts of certain Milnor invariants.

We can construct a $4$-manifold by attaching $2$-handles to a $4$-ball with framing $r$ along the components of a link in the boundary of the $4$-ball. We define a link as $r$-shake slice if there exists embedded spheres that represent the generators of the second homology of the $4$-manifold. This naturally extends $r$-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$r$-concordance for links and versions with stricter conditions on the embedded spheres that we call strongly$r$-shake slice and strongly$r$-shake concordance. We provide infinite families of links that distinguish concordance, shake concordance, and strong shake concordance. Moreover, for $r=0$ 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 $\overline{\mu }$ invariants are invariants of shake concordance. We also argue that shake concordance does not imply link homotopy.