We rederive Manolescu’s unoriented skein exact triangle for knot Floer homology over ${\mathbb{F}}_2$ combinatorially using grid diagrams, and extend it to the case with $\mathbb{Z}$ coefficients by sign refinements. Iteration of the triangle gives a cube of resolutions that converges to the knot Floer homology of an oriented link. Finally, we reestablish the homological $\sigma$–thinness of quasialternating links.

Pattern-equivariant (PE) cohomology is a well-established tool with which to interpret the Čech cohomology groups of a tiling space in a highly geometric way. We consider homology groups of PE infinite chains and establish Poincaré duality between the PE cohomology and PE homology. The Penrose kite and dart tilings are taken as our central running example; we show how through this formalism one may give highly approachable geometric descriptions of the generators of the Čech cohomology of their tiling space. These invariants are also considered in the context of rotational symmetry. Poincaré duality fails over integer coefficients for the “ePE homology groups” based upon chains which are PE with respect to orientation-preserving Euclidean motions between patches. As a result we construct a new invariant, which is of relevance to the cohomology of rotational tiling spaces. We present an efficient method of computation of the PE and ePE (co)homology groups for hierarchical tilings.

By using a notion of a geometric Dehn twist in ${♯}_k\left(S^2\times S^1\right)$, we prove that when projections of two $\mathbb{Z}$–splittings to the free factor complex are far enough from each other in the free factor complex, Dehn twist automorphisms corresponding to the $\mathbb{Z}$–splittings generate a free group of rank $2$. Moreover, every element from this free group either is conjugate to a power of one of the Dehn twists or is a fully irreducible outer automorphism of the free group. We also prove that, when the projections of $\mathbb{Z}$–splittings are sufficiently far away from each other in the intersection graph, the group generated by the Dehn twists has automorphisms that are either conjugate to Dehn twists or atoroidal fully irreducible.

Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in ${ℂ\mathbb{P}}^{n+1}$ decomposes into pairs of pants: a pair of pants is a real compact $2n$–manifold with cornered boundary obtained by removing an open regular neighborhood of $n+2$ generic complex hyperplanes from ${ℂ\mathbb{P}}^n$.

As is well-known, every compact surface of genus $g\ge 2$ decomposes into pairs of pants, and it is now natural to investigate this construction in dimension $4$. Which smooth closed $4$–manifolds decompose into pairs of pants? We address this problem here and construct many examples: we prove in particular that every finitely presented group is the fundamental group of a $4$–manifold that decomposes into pairs of pants.

We extend the theory of combinatorial link Floer homology to a class of oriented spatial graphs called transverse spatial graphs. To do this, we define the notion of a grid diagram representing a transverse spatial graph, which we call a graph grid diagram. We prove that two graph grid diagrams representing the same transverse spatial graph are related by a sequence of graph grid moves, generalizing the work of Cromwell for links. For a graph grid diagram representing a transverse spatial graph $f:G\to S^3$, we define a relatively bigraded chain complex (which is a module over a multivariable polynomial ring) and show that its homology is preserved under the graph grid moves; hence it is an invariant of the transverse spatial graph. In fact, we define both a minus and hat version. Taking the graded Euler characteristic of the homology of the hat version gives an Alexander type polynomial for the transverse spatial graph. Specifically, for each transverse spatial graph $f$, we define a balanced sutured manifold $\left(S^3\setminus f\left(G\right),\gamma \left(f\right)\right)$. We show that the graded Euler characteristic is the same as the torsion of $\left(S^3\setminus f\left(G\right),\gamma \left(f\right)\right)$ defined by S Friedl, A Juhász, and J Rasmussen.

In this article, we study the maximal length of positive Dehn twist factorizations of surface mapping classes. In connection to fundamental questions regarding the uniform topology of symplectic $4$–manifolds and Stein fillings of contact $3$–manifolds coming from the topology of supporting Lefschetz pencils and open books, we completely determine which boundary multitwists admit arbitrarily long positive Dehn twist factorizations along nonseparating curves, and which mapping class groups contain elements admitting such factorizations. Moreover, for every pair of positive integers $g$ and $n$, we tell whether or not there exist genus-$g$ Lefschetz pencils with $n$ base points, and if there are, what the maximal Euler characteristic is whenever it is bounded above. We observe that only symplectic $4$–manifolds of general type can attain arbitrarily large topology regardless of the genus and the number of base points of Lefschetz pencils on them.

We describe how to formulate Khovanov’s functor-valued invariant of tangles in the language of bordered Heegaard Floer homology. We then give an alternate construction of Lawrence Roberts’ type $D$ and type $A$ structures in Khovanov homology, and his algebra $\mathcal{ℬ}{\Gamma }_n$, in terms of Khovanov’s theory of modules over the ring $H^n$. We reprove invariance and pairing properties of Roberts’ bordered modules in this language. Along the way, we obtain an explicit generators-and-relations description of $H^n$ which may be of independent interest.

The intersection graph $M\left(i\right)$ of a generic surface $i:F\to S^3$ is the set of values which are either singularities or intersections. It is a multigraph whose edges are transverse intersections of two surfaces and whose vertices are triple intersections and branch values. $M\left(i\right)$ has an enhanced graph structure which Gui-Song Li referred to as a “daisy graph”. If $F$ is oriented, then the orientation further refines the structure of $M\left(i\right)$ into what Li called an “arrowed daisy graph”.

Li left open the question “which arrowed daisy graphs can be realized as the intersection graph of an oriented generic surface?” The main theorem of this article will answer this. I will also provide some generalizations and extensions to this theorem in Sections 4 and 5.

We show that the map on components from the space of classical long knots to the $n^{th}$ stage of its Goodwillie–Weiss embedding calculus tower is a map of monoids whose target is an abelian group and which is invariant under clasper surgery. We deduce that this map on components is a finite type-$\left(n-1\right)$ knot invariant. We compute the $E^2$–page in total degree zero for the spectral sequence converging to the components of this tower: it consists of $\mathbb{Z}$–modules of primitive chord diagrams, providing evidence for the conjecture that the tower is a universal finite-type invariant over the integers. Key to these results is the development of a group structure on the tower compatible with connected sum of knots, which in contrast with the corresponding results for the (weaker) homology tower requires novel techniques involving operad actions, evaluation maps and cosimplicial and subcubical diagrams.

This paper is devoted to discussing affine Hirsch foliations on $3$–manifolds. First, we prove that up to isotopic leaf-conjugacy, every closed orientable $3$–manifold $M$ admits zero, one or two affine Hirsch foliations. Furthermore, every case is possible.

Then we analyze the $3$–manifolds admitting two affine Hirsch foliations (we call these Hirsch manifolds). On the one hand, we construct Hirsch manifolds by using exchangeable braided links (we call such Hirsch manifolds DEBL Hirsch manifolds); on the other hand, we show that every Hirsch manifold virtually is a DEBL Hirsch manifold.

Finally, we show that for every $n\in \mathbb{N}$, there are only finitely many Hirsch manifolds with strand number $n$. Here the strand number of a Hirsch manifold $M$ is a positive integer defined by using strand numbers of braids.

For any finite subgroup $G$ of $SO\left(4\right)$, we construct a contractible $4$–manifold $C$, with an effective $G$–action on its boundary, that can be embedded in a closed $4$–manifold so that cutting $C$ out and regluing using distinct elements of $G$ will always yield distinct smooth $4$–manifolds.

We introduce an invariant for trivalent fatgraph spines of a once-bordered surface, which takes values in the first homology of the surface. This invariant is a secondary object coming from two 1–cocycles on the dual fatgraph complex, one introduced by Morita and Penner in 2008, and the other by Penner, Turaev and the author in 2013. We present an explicit formula for this invariant and investigate its properties. We also show that the mod 2 reduction of the invariant is the difference of two naturally defined spin structures on the surface.

To a Legendrian knot, one can associate an $A_{\infty }$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study this functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials.

Homological stability for sequences $G_n\to G_{n+1}\to \cdots$ of groups is often proved by studying the spectral sequence associated to the action of $G_n$ on a highly connected simplicial complex whose stabilizers are related to $G_k$ for $k<n$. When $G_n$ is the mapping class group of a manifold, suitable simplicial complexes can be made using isotopy classes of various geometric objects in the manifold. We focus on the case of surfaces and show that by using more refined geometric objects consisting of certain configurations of curves with arcs that tether these curves to the boundary, the stabilizers can be greatly simplified and consequently also the spectral sequence argument. We give a careful exposition of this program and its basic tools, then illustrate the method using braid groups before treating mapping class groups of orientable surfaces in full detail.