We define enhanced presentations of quandles via generators and relations with additional information comprising signed operations and an order structure on the set of generators. Such a presentation determines a virtual link diagram up to virtual moves. We list formal Reidemeister moves in which Tietze moves on the presented quandle are accompanied by corresponding changes to the order structure. Omitting the order structure corresponds to replacing virtual isotopy by welded isotopy.

For a hyperbolic 3–manifold $M$ with a torus boundary component, all but finitely many Dehn fillings on the torus component yield hyperbolic 3–manifolds. In this paper, we will focus on the situation where $M$ has two exceptional Dehn fillings, both of which yield toroidal manifolds. For such situation, Gordon gave an upper bound for the distance between two slopes of Dehn fillings. In particular, if $M$ is large, then the distance is at most 5. We show that this upper bound can be improved by 1 for a broad class of large manifolds.

Kuperberg [Algebr. Geom. Topol. 3 (2003) 587-591] has shown that a virtual knot diagram corresponds (up to generalized Reidemeister moves) to a unique embedding in a thickened surface of minimal genus. If a virtual knot diagram is equivalent to a classical knot diagram then this minimal surface is a sphere. Using this result and a generalised bracket polynomial, we develop methods that may determine whether a virtual knot diagram is non-classical (and hence non-trivial). As examples we show that, except for special cases, link diagrams with a single virtualization and link diagrams with a single virtual crossing are non-classical.

Given a rack $Q$ and a ring $\mathbb{A}$, one can construct a Yang–Baxter operator $c_Q:V\otimes V\to V\otimes V$ on the free $\mathbb{A}$–module $V=\mathbb{A}Q$ by setting $c_Q\left(x\otimes y\right)=y\otimes x^y$ for all $x,y\in Q$. In answer to a question initiated by D N Yetter and P J Freyd, this article classifies formal deformations of $c_Q$ in the space of Yang–Baxter operators. For the trivial rack, where $x^y=x$ for all $x,y$, one has, of course, the classical setting of $r$–matrices and quantum groups. In the general case we introduce and calculate the cohomology theory that classifies infinitesimal deformations of $c_Q$. In many cases this allows us to conclude that $c_Q$ is rigid. In the remaining cases, where infinitesimal deformations are possible, we show that higher-order obstructions are the same as in the quantum case.

Templates are branched 2–manifolds with semi-flows used to model “chaotic” hyperbolic invariant sets of flows on 3–manifolds. Knotted orbits on a template correspond to those in the original flow. Birman and Williams conjectured that for any given template the number of prime factors of the knots realized would be bounded. We prove a special case when the template is positive; the general case is now known to be false.

An orientation preserving diffeomorphism over a surface embedded in a 4–manifold is called extendable, if this diffeomorphism is a restriction of an orientation preserving diffeomorphism on this 4–manifold. In this paper, we investigate conditions for extendability of diffeomorphisms over surfaces in the complex projective plane.

An object in motivic homotopy theory is called cellular if it can be built out of motivic spheres using homotopy colimit constructions. We explore some examples and consequences of cellularity. We explain why the algebraic $K$–theory and algebraic cobordism spectra are both cellular, and prove some Künneth theorems for cellular objects.

We analyze in homological terms the homotopy fixed point spectrum of a $\mathbb{T}$–equivariant commutative $S$–algebra $R$. There is a homological homotopy fixed point spectral sequence with $E_{s,t}^2=H_{gp}^{-s}\left(\mathbb{T};H_t\left(R;{\mathbb{F}}_p\right)\right)$, converging conditionally to the continuous homology $H_{s+t}^c\left(R^{h\mathbb{T}};{\mathbb{F}}_p\right)$ of the homotopy fixed point spectrum. We show that there are Dyer–Lashof operations ${\beta }^{ϵ}{Q}^i$ acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class $x$ in the $E^{2r}$–term of the spectral sequence there are $2r$ other classes in the $E^{2r}$–term (obtained mostly by Dyer–Lashof operations on $x$) that are infinite cycles, ie survive to the $E^{\infty }$–term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra $R=THH\left(B\right)$ of many $S$–algebras, including $B=MU$, $BP$, $ku$, $ko$ and $tmf$. Similar results apply for all finite subgroups $C\subset \mathbb{T}$, and for the Tate and homotopy orbit spectral sequences. This work is part of a homological approach to calculating topological cyclic homology and algebraic $K$–theory of commutative $S$–algebras.

For line arrangements in ${\mathbb{P}}^2$ with nice combinatorics (in particular, for those which are nodal away the line at infinity), we prove that the combinatorics contains the same information as the fundamental group together with the meridianal basis of the abelianization. We consider higher dimensional analogs of the above situation. For these analogs, we give purely combinatorial complete descriptions of the following topological invariants (over an arbitrary field): the twisted homology of the complement, with arbitrary rank one coefficients; the homology of the associated Milnor fiber and Alexander cover, including monodromy actions; the coinvariants of the first higher non-trivial homotopy group of the Alexander cover, with the induced monodromy action.

We investigate the existence of an $H$–space structure on the function space, ${\mathcal{ℱ}}_{\ast }\left(X,Y,\ast \right)$, of based maps in the component of the trivial map between two pointed connected CW–complexes $X$ and $Y$. For that, we introduce the notion of $H\left(n\right)$–space and prove that we have an $H$–space structure on ${\mathcal{ℱ}}_{\ast }\left(X,Y,\ast \right)$ if $Y$ is an $H\left(n\right)$–space and $X$ is of Lusternik–Schnirelmann category less than or equal to $n$. When we consider the rational homotopy type of nilpotent finite type CW–complexes, the existence of an $H\left(n\right)$–space structure can be easily detected on the minimal model and coincides with the differential length considered by Y Kotani. When $X$ is finite, using the Haefliger model for function spaces, we can prove that the rational cohomology of ${\mathcal{ℱ}}_{\ast }\left(X,Y,\ast \right)$ is free commutative if the rational cup length of $X$ is strictly less than the differential length of $Y$, generalizing a recent result of Y Kotani.

Let $F_n$ be the free group on $n$ generators. Define ${IA}_n$ to be group of automorphisms of $F_n$ that act trivially on first homology. The Johnson homomorphism in this setting is a map from ${IA}_n$ to its abelianization. The first goal of this paper is to determine how much this map contributes to the second rational cohomology of ${IA}_n$.

A descending central series of ${IA}_n$ is given by the subgroups $K_n^{\left(i\right)}$ which act trivially on $F_n∕F_n^{\left(i+1\right)}$, the free rank $n$, degree $i$ nilpotent group. It is a conjecture of Andreadakis that $K_n^{\left(i\right)}$ is equal to the lower central series of ${IA}_n$; indeed $K_n^{\left(2\right)}$ is known to be the commutator subgroup of ${IA}_n$. We prove that the quotient group $K_n^{\left(3\right)}∕{IA}_n^{\left(3\right)}$ is finite for all $n$ and trivial for $n=3$. We also compute the rank of $K_n^{\left(2\right)}∕K_n^{\left(3\right)}$.

Let $r_m$ and $r_M$ be the least and greatest finite boundary slopes of a hyperbolic knot $K$ in $S^3$. We show that any cyclic surgery slopes of $K$ must lie in the interval $\left(r_m-1∕2,r_M+1∕2\right)$.

We prove a true bootstrapping result for convergence groups acting on a Peano continuum. We give an example of a Kleinian group $H$ which is the amalgamation of two closed hyperbolic surface groups along a simple closed curve. The limit set $\Lambda H$ is the closure of a “tree of circles" (adjacent circles meeting in pairs of points). We alter the action of $H$ on its limit set such that $H$ no longer acts as a convergence group, but the stabilizers of the circles remain unchanged, as does the action of a circle stabilizer on said circle. This is done by first separating the circles and then gluing them together backwards.

We give a necessary and sufficient condition for the addition of a collection of disjoint bypasses to a convex surface to be universally tight – namely the nonexistence of a polygonal region which we call a virtual pinwheel.

In this article we classify up to isotopy tight contact structures on Seifert manifolds over the torus with one singular fibre.

We count the number of conjugacy classes of maximal, genus $g$, surface subroups in hyperbolic 3–manifold groups. For any closed hyperbolic 3–manifold, we show that there is an upper bound on this number which grows factorially with $g$. We also give a class of closed hyperbolic 3–manifolds for which there is a lower bound of the same type.