Let $X$ be a bundle over $S^1$ with fiber a $3$–manifold $M$ and with monodromy $\phi$. Gay and Kirby showed that if $\phi$ fixes a genus $g$ Heegaard splitting of $M$ then $X$ has a genus $6g+1$ trisection. Genus $3g+1$ trisections have been found in certain special cases, such as the case where $\phi$ is trivial, and it is known that trisections of genus lower than $3g+1$ cannot exist in general. We generalize these results to prove that there exists a trisection of genus $3g+1$ whenever $\phi$ fixes a genus $g$ Heegaard surface of $M$. This means that $\phi$ can be nontrivial, and can preserve or switch the two handlebodies of the Heegaard splitting. We additionally describe an algorithm to draw a diagram for such a trisection given a Heegaard diagram for $M$ and a description of $\phi$.

We show, at the prime 2, that the Picard group of invertible modules over $E_n^{h{C}_2}$ is cyclic. Here, $E_n$ is the height $n$ Lubin–Tate spectrum and its $C_2$–action is induced from the formal inverse of its associated formal group law. We further show that $E_n^{h{C}_2}$ is Gross–Hopkins self-dual and determine the exact shift. Our results generalize the well-known results when $n=1$.

A knot $\kappa$ in $S^3$ is persistently foliar if, for each nontrivial boundary slope, there is a cooriented taut foliation meeting the boundary of the knot complement transversely in a foliation by curves of that slope. For rational slopes, these foliations may be capped off by disks to obtain a cooriented taut foliation in every manifold obtained by nontrivial Dehn surgery on that knot. We show that any composite knot with a persistently foliar summand is persistently foliar and that any nontrivial connected sum of fibered knots is persistently foliar. As an application, it follows that any composite knot in which each of two summands is fibered or at least one summand is nontorus alternating or Montesinos is persistently foliar.

We note that, in constructing foliations in the complements of fibered summands, we build branched surfaces whose complementary regions agree with those of Gabai’s product disk decompositions, except for the one containing the boundary of the knot complement. It is this boundary region which provides for persistence.

Let $C_2$ denote the cyclic group of order $2$. Given a manifold with a $C_2$–action, we can consider its equivariant Bredon $RO\left(C_2\right)$–graded cohomology. We develop a theory of fundamental classes for equivariant submanifolds in $RO\left(C_2\right)$–graded cohomology with constant $\mathbb{Z}∕2$–coefficients. We show the cohomology of any $C_2$–surface is generated by fundamental classes, and these classes can be used to easily compute the ring structure. To define fundamental classes we are led to study the cohomology of Thom spaces of equivariant vector bundles. In general, the cohomology of the Thom space is not just a shift of the cohomology of the base space, but we show there are still elements that act as Thom classes, and cupping with these classes gives an isomorphism within a certain range.

We construct the augmentation representation. It is a representation of the fundamental group of the link complement associated to an augmentation of the framed cord algebra. This construction connects representations of two link invariants of different types. We also study properties of the augmentation representation.

A continuous fibration of ${ℝ}^3$ by oriented lines is given by a unit vector field $V:{ℝ}^3\to S^2$ for which all of the integral curves are oriented lines. A line fibration is called skew if no two fibers are parallel. Skew fibrations have been the focus of recent study, in part due to their close relationships with great circle fibrations of $S^3$ and with tight contact structures on ${ℝ}^3$. Both geometric and topological classifications of the space of skew fibrations have appeared; these classifications rely on certain rigid geometric properties exhibited by skew fibrations. Here we study these properties for line fibrations which are not necessarily skew, and we offer some partial answers to the question: in what sense do nonskew fibrations look and behave like skew fibrations? We develop and utilize a technique, called the parallel plane pushoff, for studying nonskew fibrations. In addition, we summarize the known relationship between line fibrations and contact structures, and we extend these results to give a complete correspondence. Finally, we develop a technique for generating nonskew fibrations and offer a number of examples. All results (except where otherwise stated) hold in the topological category.

We determine the asymptotic growth rate of the diameter of the random hyperbolic surfaces constructed by Brooks and Makover (J. Differential Geom. 68 (2004) 121–157). This model consists of a uniform gluing of $2n$ hyperbolic ideal triangles along their sides followed by a compactification to get a random hyperbolic surface of genus roughly $\frac{n}{2}$. We show that the diameter of those random surfaces is asymptotic to $2logn$ in probability as $n\to \infty$.

The Kauffman bracket skein module $\mathcal{𝒮}\left(M\right)$ of a $3$–manifold $M$ is a $\mathbb{Q}\left(A\right)$–vector space spanned by links in $M$ modulo the so-called Kauffman relations. For any closed oriented surface $\mathrm{\Sigma }$ we provide an explicit spanning family for the skein modules $\mathcal{𝒮}\left(\mathrm{\Sigma }\times S^1\right)$. Combined with earlier work of Gilmer and Masbaum (Proc. Amer. Math. Soc. 147 (2019) 4091–4106), we answer their question about the dimension of $\mathcal{𝒮}\left(\mathrm{\Sigma }\times S^1\right)$ being $2^{2g+1}+2g-1$.

We construct explicit examples of geodesics in the mapping class group and show that the shadow of a geodesic in the mapping class group to the curve graph does not have to be a uniform quality quasigeodesic. We also show that the quasiaxis of a pseudo-Anosov element of the mapping class group may not have the strongly contracting property. Specifically, we show that, after choosing a generating set carefully, one can find a pseudo-Anosov homeomorphism $\varphi$, a sequence of points $w_k$ and a sequence of radii $r_k$ such that the ball $B\left(w_k,r_k\right)$ is disjoint from a quasiaxis ${\mathit{a}}_{\varphi }$ of $\varphi$, but, for any projection from the mapping class group to ${\mathit{a}}_{\varphi }$, the diameter of the image of $B\left(w_k,r_k\right)$ grows like $log\left(r_k\right)$.

Sutured monopole and instanton Floer homologies were introduced by Kronheimer and Mrowka (J. Differential Geom. 84 (2010) 301–364). We construct cobordism maps in these two theories and prove that such cobordism maps are functorial under the composition of cobordisms and have a duality property. Along with the construction, we also construct gluing maps in sutured monopole and instanton theories, which are of independent interest.

A cobordism between links in thickened surfaces consists of a surface $S$ and a $3$–manifold $M$ with $S$ properly embedded in $M\times I$. We show that there exist links in thickened surfaces such that if $\left(S,M\right)$ is a cobordism between them in which $S$ is simple, then $M$ must be complex. That is, there are cases in which low complexity of the surface does not imply low complexity of the $3$–manifold.

Specifically, we show that there exist concordant links in thickened surfaces between which a concordance can only be realized by passing through thickenings of higher-genus surfaces. We exhibit an infinite family of such links that are detected by an elementary method and other families of links that are not detectable in this way. We investigate an augmented version of Khovanov homology, and use it to detect these families. Such links provide counterexamples to an analogue of the slice–ribbon conjecture.

We prove that alternating links with two totally geodesic checkerboard surfaces are three links with projection the $1$–skeleton of the octahedron, the cuboctahedron and the icosidodecahedron. Then we characterize these links as right-angled completely realizable links and show that all hyperbolic weaving knots with two exceptions have both checkerboard surfaces not totally geodesic.

We show that any open orientable surface $S$ can be properly embedded in ${ℍ}^2\times ℝ$ as an area-minimizing surface.

We use purely topological tools to construct several infinite families of hyperbolic links inthe $3$–sphere that satisfy the Turaev–Viro invariant volume conjecture posed by Chen and Yang. To show that our links satisfy the volume conjecture, we prove that each has complement homeomorphic to the complement of a fundamental shadow link. These are links in connected sums of copies of $S^2\times S^1$ for which the conjecture is known due to Belletti, Detcherry, Kalfagianni and Yang. Our methods also verify the conjecture for several hyperbolic links with crossing number less than twelve. In addition, we show that every link in the $3$–sphere is a sublink of a link that satisfies the conjecture.

As an application of our results, we extend the class of known examples that satisfy the AMU conjecture on quantum representations of surface mapping class groups. For example, we give explicit elements in the mapping class group of a genus $g$ surface with four boundary components for any $g$. For this, we use techniques developed by Detcherry and Kalfagianni which relate the Turaev–Viro invariant volume conjecture to the AMU conjecture.

We describe presentations of the Roger–Yang generalized skein algebras for punctured spheres with an arbitrary number of punctures. This skein algebra is a quantization of the decorated Teichmüller space and generalizes the construction of the Kauffman bracket skein algebra. We also obtain a new interpretation of the homogeneous coordinate ring of the Grassmannian of planes in terms of skein theory.