We use elementary skein theory to prove a version of a result of Stylianakis (Stylianakis, The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere, arXiv:1511.02912) who showed that under mild restrictions on m and n, the normal closure of the mth power of a half-twist has infinite index in the mapping class group of a sphere with 2n punctures.

A multicurve${\mathcal{C}}$ in a closed orientable surface Sg of genus g is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. A left-handed Dehn twist$t_{\mathcal{C}}$about${\mathcal{C}}$ is the product of left-handed Dehn twists about the individual curves in ${\mathcal{C}}$. In this paper, we derive necessary and sufficient conditions for the existence of a root of $t_{\mathcal{C}}$ in the mapping class group Mod(Sg). Using these conditions, we obtain combinatorial data that correspond to roots, and use it to determine upper bounds on the degree of a root. As an application of our theory, we classify all such roots up to conjugacy in Mod(S4). Finally, we establish that no such root can lie in the level m congruence subgroup of Mod(Sg), for m ≥ 3.

We advocate the use of cluster algebras and their $y$-variables in the study of hyperbolic 3-manifolds. We study hyperbolic structures on the mapping tori of pseudo-Anosov mapping classes of punctured surfaces, and show that cluster $y$-variables naturally give the solutions of the edge-gluing conditions of ideal tetrahedra. We also comment on the completeness of hyperbolic structures.

We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involutive correction terms in the case that all of the summands have the same orientation. Using these calculations, we give a new proof of the existence of an infinite-rank subgroup in the three-dimensional homology cobordism group.

We study cluster categories arising from marked surfaces (with punctures and non-empty boundaries). By constructing skewed-gentle algebras, we show that there is a bijection between tagged curves and string objects. Applications include interpreting dimensions of $\operatorname{Ext}^{1}$ as intersection numbers of tagged curves and Auslander–Reiten translation as tagged rotation. An important consequence is that the cluster(-tilting) exchange graphs of such cluster categories are connected.

We show that Cannon–Thurston maps exist for degenerate free groups without parabolics, that is, for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon–Thurston maps for surface groups, we show that Cannon–Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show that point pre-images under Cannon–Thurston maps for degenerate free groups without parabolics correspond to endpoints of leaves of an ending lamination in the Masur domain, whenever a point has more than one pre-image. This proves a conjecture of Otal. We also prove a similar result for point pre-images under Cannon–Thurston maps for arbitrary finitely generated Kleinian groups without parabolics.

We show how the quantum trace map of Bonahon and Wong can be constructed in a natural way using the skein algebra of Muller, which is an extension of the Kauffman bracket skein algebra of surfaces. We also show that the quantum Teichmüller space of a marked surface, defined by Chekhov–Fock (and Kashaev) in an abstract way, can be realized as a concrete subalgebra of the skew field of the skein algebra.

We present a graph manifold analog of the Jankins–Neumann classification of Seifert fibered spaces over $S^{2}$ admitting taut foliations, providing a finite recursive formula to compute the L-space Dehn-filling interval for any graph manifold with torus boundary. As an application of a generalization of this result to Floer simple manifolds, we compute the L-space interval for any cable of a Floer simple knot complement in a closed three-manifold in terms of the original L-space interval, recovering a result of Hedden and Hom as a special case.

A killer of a group $G$ is an element that normally generates $G$. We show that the group of a cable knot contains infinitely many killers such that no two lie in the same automorphic orbit.

In this paper, we determine the group of contact transformations modulo contact isotopies for Legendrian circle bundles over closed surfaces of non-positive Euler characteristic. These results extend and correct those presented by the first author in a former work. The main ingredient we use is connectedness of certain spaces of embeddings of surfaces into contact 3-manifolds. This connectedness question is also studied for itself with a number of (hopefully instructive) examples.

Excluding four exceptional cases, we determine the asphericity of the relative presentation for m ⩾ 2. If H = ⟨g, h⟩ ⩽ G, then the exceptional cases occur when H is isomorphic to C5 or C6.

The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering links and Alexander invariants. As an application, we completely determine the splitting numbers of links with nine or fewer crossings. Also, with these techniques, we either reprove or improve upon the lower bounds for splitting numbers of links computed by Batson and Seed using Khovanov homology.

Every finite group $G$ acts on some nonorientable unbordered surfaces. The minimal topological genus of those surfaces is called the symmetric crosscap number of $G$. It is known that 3 is not the symmetric crosscap number of any group but it remains unknown whether there are other such values, called gaps. In this paper we obtain group presentations which allow one to find the actions realizing the symmetric crosscap number of groups of each group of order less than or equal to 63.

We exhibit a knot $P$ in the solid torus, representing a generator of first homology, such that for any knot $K$ in the 3-sphere, the satellite knot with pattern $P$ and companion $K$ is not smoothly slice in any homology 4-ball. As a consequence, we obtain a knot in a homology 3-sphere that does not bound a piecewise-linear disk in any homology 4-ball.

In this paper we discuss a relationship between the spectral asymmetry and the surface symmetry. More precisely, we show that for every automorphism of a Hurwitz surface with the automorphism group $\text{PSL}(2,\mathbb{F}_{q})$, the $\unicode[STIX]{x1D702}$-invariant of the corresponding mapping torus vanishes if $q$ is sufficiently large.

We generalize to higher dimensions the Bavard–Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex $d$-dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams. We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The Alexandrov–Fenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting cone-manifold are equal to $\unicode[STIX]{x1D70B}/2$.

We give a method for constructing a Legendrian representative of a knot in $S^{3}$ which realizes its maximal Thurston–Bennequin number under a certain condition. The method utilizes Stein handle decompositions of $D^{4}$ , and the resulting Legendrian representative is often very complicated (relative to the complexity of the topological knot type). As an application, we construct infinitely many knots in $S^{3}$ each of which yields a reducible 3-manifold by a Legendrian surgery in the standard tight contact structure. This disproves a conjecture of Lidman and Sivek.

In this paper we use the Alexander ideals of groups to solve the isomorphism problem for the Baumslag–Solitar groups and a family of parafree groups introduced by Baumslag and Cleary.

The Gehring–Martin–Tan inequality for two-generator subgroups of $\text{PSL}(2,\mathbb{C})$ is one of the best known discreteness conditions. A Kleinian group $G$ is called a Gehring–Martin–Tan group if the equality holds for the group $G$. We give a method for constructing Gehring–Martin–Tan groups with a generator of order four and present some examples. These groups arise as groups of finite-volume hyperbolic 3-orbifolds.

In this paper we propose and discuss implications of a general conjecture that there is a natural action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K\subset S^{3}$. We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$). As the main application of the conjecture, we construct three-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev. We also deduce some new properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies that the colored Jones polynomials of $K$ satisfy an inhomogeneous recursion relation.