Archdeacon and Grable (1995) proved that the genus of the random graph $G\in {\mathcal{G}}_{n,p}$ is almost surely close to $pn^{2}/12$ if $p=p(n)\geqslant 3(\ln n)^{2}n^{-1/2}$. In this paper we prove an analogous result for random bipartite graphs in ${\mathcal{G}}_{n_{1},n_{2},p}$. If $n_{1}\geqslant n_{2}\gg 1$, phase transitions occur for every positive integer $i$ when $p=\unicode[STIX]{x1D6E9}((n_{1}n_{2})^{-i/(2i+1)})$. A different behaviour is exhibited when one of the bipartite parts has constant size, i.e., $n_{1}\gg 1$ and $n_{2}$ is a constant. In that case, phase transitions occur when $p=\unicode[STIX]{x1D6E9}(n_{1}^{-1/2})$ and when $p=\unicode[STIX]{x1D6E9}(n_{1}^{-1/3})$.

We give simple homological conditions for a rational homology 3-sphere $Y$ to have infinite order in the rational homology cobordism group $\unicode[STIX]{x1D6E9}_{\mathbb{Q}}^{3}$, and for a collection of rational homology spheres to be linearly independent. These translate immediately to statements about knot concordance when $Y$ is the branched double cover of a knot, recovering some results of Livingston and Naik. The statements depend only on the homology groups of the 3-manifolds, but are proven through an analysis of correction terms and their behavior under connected sums.

Under certain assumptions on CAT(0) spaces, we show that the geodesic flow is topologically mixing. In particular, the Bowen–Margulis’ measure finiteness assumption used by Ricks [Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces. Ergod. Th. & Dynam. Sys. 37 (2017), 939–970] is removed. We also construct examples of CAT(0) spaces that do not admit finite Bowen–Margulis measure.

We give a complete computation of the Bieri–Neumann–Strebel–Renz invariants Σm(Hn) of the Houghton groups Hn. Partial results were previously obtained by the author, with a conjecture about the full picture, which we now confirm. The proof involves covering relevant subcomplexes of an associated CAT (0) cube complex by their intersections with certain locally convex subcomplexes, and then applying a strong form of the Nerve Lemma. A consequence of the full computation is that for each 1 ≤ m ≤ n − 1, Hn admits a map onto ℤ whose kernel is of type Fm−1 but not Fm; moreover, no such kernel is ever of type Fn−1.

We analyse the asymptotic extremal growth rate of the Betti numbers of clique complexes of graphs on n vertices not containing a fixed forbidden induced subgraph H.

In particular, we prove a theorem of the alternative: for any H the growth rate achieves exactly one of five possible exponentials, that is, independent of the field of coefficients, the nth root of the maximal total Betti number over n-vertex graphs with no induced copy of H has a limit, as n tends to infinity, and, ranging over all H, exactly five different limits are attained.

For the interesting case where H is the 4-cycle, the above limit is 1, and we prove a superpolynomial upper bound.

In this paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to the computation of the splitting number. Finally, we use the slice-torus link invariants and the Whitehead doubling to define new strong concordance invariants for links, which are proven to be independent of the corresponding slice-torus link invariant.

We prove that for $n\geqslant 4$, every knot has infinitely many conjugacy classes of $n$-braid representatives if and only if it has one admitting an exchange move.

We construct infinitely many compact, smooth 4-manifolds which are homotopy equivalent to $S^{2}$ but do not admit a spine (that is, a piecewise linear embedding of $S^{2}$ that realizes the homotopy equivalence). This is the remaining case in the existence problem for codimension-2 spines in simply connected manifolds. The obstruction comes from the Heegaard Floer $d$ invariants.

We define a metric filtration of the Gordian graph by an infinite family of 1-dense subgraphs. The nth subgraph of this family is generated by all knots whose fundamental groups surject to a symmetric group with parameter at least n, where all meridians are mapped to transpositions. Incidentally, we verify the Meridional Rank Conjecture for a family of knots with unknotting number one yet arbitrarily high bridge number.

We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to Stoffregen’s connected Seiberg–Witten Floer homology. We use this invariant to study the structure of the homology cobordism group and, along the way, compute the involutive correction terms $\bar{d}$ and $\text{}\underline{d}$ for certain families of three-manifolds.

We study 2-generated subgroups $\langle f,g\rangle <\operatorname{Homeo}^{+}(I)$ such that $\langle f^{2},g^{2}\rangle$ is isomorphic to Thompson’s group $F$, and such that the supports of $f$ and $g$ form a chain of two intervals. We show that this class contains uncountably many isomorphism types. These include examples with non-abelian free subgroups, examples which do not admit faithful actions by $C^{2}$ diffeomorphisms on 1-manifolds, examples which do not admit faithful actions by $PL$ homeomorphisms on an interval, and examples which are not finitely presented. We thus answer questions due to Brin. We also show that many relatively uncomplicated groups of homeomorphisms can have very complicated square roots, thus establishing the behavior of square roots of $F$ as part of a general phenomenon among subgroups of $\operatorname{Homeo}^{+}(I)$.

An embedding of a metric graph $(G,d)$ on a closed hyperbolic surface is essential if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_{e}(G)$ of $(G,d)$ is the lowest genus of a surface on which such an embedding is possible. We establish a formula to compute $g_{e}(G)$ and show that, for every integer $g\geq g_{e}(G)$, there is an embedding of $(G,d)$ (possibly after a rescaling of $d$) on a surface of genus $g$. Next, we study minimal embeddings where each complementary region has Euler characteristic $-1$. The maximum essential genus $g_{e}^{\max }(G)$ of $(G,d)$ is the largest genus of a surface on which the graph is minimally embedded. We describe a method for an essential embedding of $(G,d)$, where $g_{e}(G)$ and $g_{e}^{\max }(G)$ are realised.

We resolve parts (A) and (B) of Problem 1.100 from Kirby’s list [Problems in low-dimensional topology, in Geometric topology, AMS/IP Studies in Advanced Mathematics, vol. 2 (American Mathematical Society, Providence, RI, 1997), 35–473] by showing that many nontrivial links arise as cross-sections of unknotted holomorphic disks in the four-ball. The techniques can be used to produce unknotted ribbon surfaces with prescribed cross-sections, including unknotted Lagrangian disks with nontrivial cross-sections.

We describe in this article the dynamics of a one-parameter family of affine interval exchange transformations. This amounts to studying the directional foliations of a particular dilatation surface introduced in Duryev et al [Affine surfaces and their Veech groups. Preprint, 2016, arXiv:1609.02130], the Disco surface. We show that this family displays various dynamical behaviours: it is generically dynamically trivial but for a Cantor set of parameters the leaves of the foliations accumulate to a (transversely) Cantor set. This study is achieved through analysis of the dynamics of the Veech group of this surface combined with a modified version of Rauzy induction in the context of affine interval exchange transformations.

We present a new test for studying asphericity and diagrammatic reducibility of group presentations. Our test can be applied to prove diagrammatic reducibility in cases where the classical weight test fails. We use this criterion to generalize results of J. Howie and S.M. Gersten on asphericity of LOTs and of Adian presentations, and derive new results on solvability of equations over groups. We also use our methods to investigate a conjecture of S.V. Ivanov related to Kaplansky's problem on zero divisors: we strengthen Ivanov's result for locally indicable groups and prove a weak version of the conjecture.

It is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of the braid index of an alternating link, Yamada showed that the minimum number of Seifert circles over all regular projections of a link equals the braid index. Thus one may conjecture that the number of Seifert circles in a reduced alternating diagram of the link equals the braid index of the link, but this turns out to be false. In this paper we prove the next best thing that one could hope for: we characterise exactly those alternating links for which their braid indices equal the numbers of Seifert circles in their corresponding reduced alternating link diagrams. More specifically, we prove that if D is a reduced alternating link diagram of an alternating link L, then b(L), the braid index of L, equals the number of Seifert circles in D if and only if GS(D) contains no edges of weight one. Here GS(D), called the Seifert graph of D, is an edge weighted simple graph obtained from D by identifying each Seifert circle of D as a vertex of GS(D) such that two vertices in GS(D) are connected by an edge if and only if the two corresponding Seifert circles share crossings between them in D and that the weight of the edge is the number of crossings between the two Seifert circles. This result is partly based on the well-known MFW inequality, which states that the a-span of the HOMFLY polynomial of L is a lower bound of 2b(L)−2, as well as the result of Yamada relating the minimum number of Seifert circles over all link diagrams of L to b(L).

The profinite completion of the fundamental group of a closed, orientable $3$-manifold determines the Kneser–Milnor decomposition. If $M$ is irreducible, then the profinite completion determines the Jaco–Shalen–Johannson decomposition of $M$.

We introduce a new method for computing triply graded link homology, which is particularly well adapted to torus links. Our main application is to the $(n,n)$-torus links, for which we give an exact answer for all $n$. In several cases, our computations verify conjectures of Gorsky et al. relating homology of torus links with Hilbert schemes.

Fibonacci anyons are attractive for use in topological quantum computation because any unitary transformation of their state space can be approximated arbitrarily accurately by braiding. However, there is no known braid that entangles two qubits without leaving the space spanned by the two qubits. In other words, there is no known ‘leakage-free’ entangling gate made by braiding. In this paper, we provide a remedy to this problem by supplementing braiding with measurement operations in order to produce an exact controlled rotation gate on two qubits.