Narrow Results

Athreya et al. [Lattice point asymptotics and volume growth on Teichmüller space, Duke Math. J.161 (2012) 1055–1111] have shown the number of mapping class group lattice points intersecting a closed ball of radius $R$ in Teichmüller space is asymptotic to $e^{hR}$, where $h$ is the dimension of the Teichmüller space. In contrast we show the number of Dehn twist lattice points intersecting a closed ball of radius $R$ is coarsely asymptotic to $e^{\frac{h}{2}R}$. Moreover, we show the number multi-twist lattice points intersecting a closed ball of radius $R$ grows coarsely at least at the rate of $R\cdot e^{\frac{h}{2}R}$.

For $g\ge 2$, let $\mathrm{\text{Mod}}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g$. In this paper, we obtain necessary and sufficient conditions under which a given pseudo-periodic mapping class can be a root of another up to conjugacy. Using this characterization, the canonical decomposition of (non-periodic) mapping classes, and some known algorithms, we give an algorithm for determining the conjugacy classes of roots of arbitrary mapping classes. Furthermore, we derive realizable bounds on the degrees of roots of pseudo-periodic mapping classes in $\mathrm{\text{Mod}}(S_g)$, the Torelli group, the level-$m$ subgroup of $\mathrm{\text{Mod}}(S_g)$, and the commutator subgroup of $\mathrm{\text{Mod}}(S_2)$. In particular, we show that the highest possible (realizable) degree of a root of a pseudo-periodic mapping class $F$ is $3q(F)(g+1)(g+2)$, where $q(F)$ is a unique positive integer associated with the conjugacy class of $F$. Moreover, this bound is realized by a root of a power of a Dehn twist about a separating curve of genus $[g/2]$ in $S_g$, where $g\equiv 0,9(mod12)$. Finally, for $g\ge 3$, we show that any pseudo-periodic mapping class having a nontrivial periodic component that is not the hyperelliptic involution, normally generates $\mathrm{\text{Mod}}(S_g)$. Consequently, we establish that $\mathrm{\text{Mod}}(S_g)$ is normally generated by a root of bounding pair map or a root of a nontrivial power of a Dehn twist.

We use open book representations of contact 3-manifolds to compute the cylindrical contact homology of a Stein-fillable contact 3-manifold represented by the open book whose monodromy is a positive Dehn twist on a torus with boundary.

For a surface F bounding a handlebody H, we look at simple closed curves on F which intersect every disk in the handlebody, at least n times (called n-closed curves). We give a finite criterion for a curve to be n-closed. Using this, we derive a sufficiency condition for a Heegaard splitting to be strongly irreducible. We then look at further intersection properties of curves with disk families in H. In particular, we look at the effects of Dehn twists on n-closed curves, and using a finite fixed disk collection as a coordinate system, give heuristics and a counting formula for measuring the number of intersections of the resulting curves, with disks in H. In a certain instance, this yields a partial "grading" on the Dehn twist quandle with respect to the degree of n-closedness.

We find two new classes of virtually fibered classic Montesinos links of type .

In view of the self-linking invariant, the number |K| of framed knots in S³ with given underlying knot K is infinite. In fact, the second author previously defined affine self-linking invariants and used them to show that |K| is infinite for every knot in an orientable manifold unless the manifold contains a connected sum factor of S¹ × S²; the knot K need not be zero-homologous and the manifold is not required to be compact. We show that when M is orientable, the number |K| is infinite unless K intersects a nonseparating sphere at exactly one point, in which case |K| = 2; the existence of a nonseparating sphere implies that M contains a connected sum factor of S¹ × S². For knots in nonorientable manifolds we show that if |K| is finite, then K is disorienting, or there is an orientation-preserving isotopy of the knot to itself which changes the orientation of its normal bundle, or it intersects some embedded S² or ℝP² at exactly one point, or it intersects some embedded S² at exactly two points in such a way that a closed curve consisting of an arc in K between the intersection points and an arc in S² is disorienting.

We give new upper bounds on the stable commutator lengths of Dehn twists along separating curves in the mapping class group of a closed oriented surface. The estimates of these upper bounds are $O(1/g)$, where $g$ is the genus of the surface.

Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We investigate the problem of roots of a Dehn twist $t_c$ about a nonseparating circle $c$ in the mapping class group ${ℳ}(N_g)$ of a nonorientable surface $N_g$ of genus $g$. We explore the existence of roots and, following the work of McCullough, Rajeevsarathy and Monden, give a simple arithmetic description of their conjugacy classes. We also study roots of maximal degree and prove that if we fix an odd integer $n>1$, then for each sufficiently large $g$, $t_c$ has a root of degree $n$ in ${ℳ}(N_g)$. Moreover, for any possible degree $n$, we provide explicit expressions for a particular type of roots of Dehn twists about nonseparating circles in $N_g$.

We investigate the properties of certain operators that were used in previous papers to create geometric and algebraic intersection number functions. Using these operators we show how to construct a general intersection theory and then we investigate properties of certain polynomials that arise naturally in this context. We indicate connections to twin primes.

For any unoriented loop on a compact connected oriented surface with one boundary component, we introduce a generalized Dehn twist along the loop as a certain automorphism of the completed group ring of the fundamental group of the surface. If the loop is simple, this corresponds to the right-handed Dehn twist, and in particular is realized as a diffeomorphism of the surface. We investigate the case where the loop has a single transverse double point, and show that in this case the generalized Dehn twist is not realized as a diffeomorphism.