Journal of Knot Theory and Its Ramifications

We present a large family of knots for which the Rasmussen $s$-invariants of arbitrary satellites do not detect sliceness. This answers a question of Hedden. The proof hinges on work of Kronheimer–Mrowka and Cochran–Harvey–Horn.

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 prove a conjecture of Riley on $\mathrm{\text{SL}}(2,ℝ)$ representations of 2-bridge knot groups. As a consequence we show that if $K$ is a 2-bridge knot with non-zero signature, then the fundamental group of the $n$-fold cyclic branched cover of $K$ is left-orderable for $n$ sufficiently large.

A well-known conjecture states that for any $l$-component link $L$ in $S^3$, the rank of the knot Floer homology of $L$ (over any field) is less than or equal to $2^{l-1}$ times the rank of the reduced Khovanov homology of $L$. In this paper, we describe a framework that might be used to prove this conjecture. We construct a modified version of Khovanov homology for links with multiple basepoints and show that it mimics the behavior of knot Floer homology. We also introduce a new spectral sequence converging to knot Floer homology whose $E_1$ page is conjecturally isomorphic to our new version of Khovanov homology; this would prove that the conjecture stated above holds over the field ${\mathbb{Z}}_2$.

This paper investigates which smooth manifolds arise as quotients (orbit spaces) of flows of vector fields. Such quotient maps were already known to be surjective on fundamental groups, but this paper shows that every epimorphism of countably presented groups is induced by the quotient map of some flow, and that higher homology can also be controlled. Manifolds of fixed dimension arising as quotients of flows on Euclidean space realize all even (and some odd) intersection pairings, and all homotopy spheres of dimension at least two arise in this manner. Most Euclidean spaces of dimensions five and higher have families of topologically equivalent but smoothly inequivalent flows with quotient homeomorphic to a manifold with flexibly chosen homology. For $m\ge 2r>2$, there is a topological flow on (ℝ^2r+1 − 8 points) × ℝ^m that is unsmoothable, although smoothable near each orbit, with quotient an unsmoothable topological manifold.

This paper contains two remarks about the application of the $d$-invariant in Heegaard Floer homology and Donaldson's diagonalization theorem to knot theory. The first is the equivalence of two obstructions they give to a 2-bridge knot being smoothly slice. The second carries out a suggestion by Stefan Friedl to replace the use of Heegaard Floer homology by Donaldson's theorem in the proof of the main result of [J. E. Greene, Lattices, graphs, and Conway mutation, Invent. Math. 192(3) (2013) 717–750] concerning Conway mutation of alternating links.

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 give a new proof that the Levine–Tristram signatures of a link give lower bounds for the minimal sum of the genera of a collection of oriented, locally flat, disjointly embedded surfaces that the link can bound in the 4-ball. We call this minimal sum the 4-genus of the link. We also extend a theorem of Cochran, Friedl and Teichner to show that the 4-genus of a link does not increase under infection by a string link, which is a generalized satellite construction, provided that certain homotopy triviality conditions hold on the axis curves, and that enough Milnor's $\overline{\mu }$-invariants of the closure of the infection string link vanish. We construct knots for which the combination of the two results determines the 4-genus.

We exhibit the traceless SU(2) character variety of a 6-punctured 2-sphere as a 2-fold branched cover of $ℂP^3$, branched over the singular Kummer surface, with the branch locus in $R(S^2,6)$ corresponding to the binary dihedral representations. This follows from an analysis of the map induced on SU(2) character varieties by the 2-fold branched cover $F_{n-1}\to S^2$ branched over $2n$ points, combined with the theorem of Narasimhan–Ramanan which identifies $R(F_2)$ with $ℂP^3$. The singular points of $R(S^2,6)$ correspond to abelian representations, and we prove that each has a neighborhood in $R(S^2,6)$ homeomorphic to a cone on $S^2\times S^3$.

In this paper, we study the notion of critical dimension of random simplicial complexes in the general multi-parameter model described in [Random simplicial complexes, preprint (2014), arXiv:1412.5805; Large random simplicial complexes, I, preprint (2015), arXiv:1503.06285; Large random simplical complexes, II, preprint (2015), arXiv:1509.04837]. This model includes as special cases the Linial–Meshulam–Wallach model [Homological connectivity of random 2-complexes, Combinatorica 26 (2006) 475–487; Homological connectivity of random $K$-complexes, Random Struct. Alogrithms 34 (2009) 408–417.] as well as the clique complexes of random graphs. We characterize the concept of critical dimension in terms of various geometric and topological properties of random simplicial complexes such as their Betti numbers, the fundamental group, the size of minimal cycles and the degrees of simplexes. We mention in the text a few interesting open questions.

This paper introduces a number of new intrinsically 3-linked graphs through five new constructions. We prove that intrinsic 3-linkedness is not preserved by $\text{Y}\nabla$ moves. We will see that the graph $M$, which is obtained through a $\text{Y}\nabla$ move on ${(PG)}_{\ast }^{\ast }(PG)$, is not intrinsically 3-linked.

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.

In the early 2000's Cochran and Harvey introduced non-commutative Alexander polynomials for 3-manifolds. Their degrees give strong lower bounds on the Thurston norm. In this paper, we make the case that the vanishing of a certain Novikov–Sikorav homology module is the correct notion of a monic non-commutative Alexander polynomial. Furthermore we will use the opportunity to give new proofs of several statements about Novikov–Sikorav homology in the three-dimensional context.

Let $X$ and $Y$ be the complementary regions of a closed hypersurface $M$ in $S^4=X{\cup }_{M}Y$. We use the Massey product structure in $H^{\ast }(M;\mathbb{Z})$ to limit the possibilities for $\chi (X)$ and $\chi (Y)$. We show also that if ${\pi }_1(X)\ne 1$ then it may be modified by a 2-knot satellite construction, while if $\chi (X)\le 1$ and ${\pi }_1(X)$ is abelian then ${\beta }_1(M)\le 4$ or ${\beta }_1(M)=6$. Finally we use TOP surgery to propose a characterization of the simplest embeddings of $F\times S^1$.

In this survey paper, we discuss several different knot concordance invariants coming from the Heegaard Floer homology package of Ozsváth and Szabó. Along the way, we prove that if two knots are concordant, then their knot Floer complexes satisfy a certain type of stable equivalence.

We give necessary and sufficient conditions for a given polynomial to be a plucking polynomial of a rooted tree. We discuss the fact that different rooted trees can have the same polynomial.

The purpose of this paper is to give a complete description of the Cohn localization of the augmentation map $Z[G]\to Z$ when $G$ is any finite group.

The A-B slice problem, a reformulation of the four-dimensional topological surgery conjecture for free groups, is shown to admit a link-homotopy${}^{+}$ solution. The proof relies on geometric applications of the group-theoretic $2$-Engel relation. Implications for the surgery conjecture are discussed.