Related papers: The maximal number of exceptional Dehn surgeries

We give an upper bound for the number of compact essential orientable non-isotopic surfaces, with Euler characteristic at least some constant $\chi$, properly embedded in a finite-volume hyperbolic 3-manifold $M$, closed or cusped. This…

We prove that for any integer $n$ there exist infinitely many different knots in $S^3$ such that $n$-surgery on those knots yields the same 3-manifold. In particular, when $|n|=1$ homology spheres arise from these surgeries. This answers…

The maximum cut problem for a quintic del Pezzo surface ${\rm Bl}_{4}(\mathbb{P}^2)$ asks: Among all partitions of the 10 exceptional curves into two disjoint sets, what is the largest possible number of pairwise intersections? In this…

We study the maximum number of straight-line segments connecting $n$ points in convex position in the plane, so that each segment intersects at most $k$ others. This question can also be framed as the maximum number of edges of an outer…

A graph $G$ is $H$-covered by some given graph $H$ if each vertex in $G$ is contained in a copy of $H$. In this note, we give the maximum number of independent sets of size $t\ge 3$ in $K_n$-covered graphs of size $N\ge n+t-1$ and determine…

In this article, we provide bounds on systoles associated to a holomorphic $1$-form $\omega$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on…

In this article, for any $n\geq 4$ we construct a sequence of compact hyperbolic $n$-manifolds $\{M_i\}$ with number of systoles at least as $\mathrm{vol}(M_i)^{1+\frac{1}{3n(n+1)}-\epsilon}$ for any $\epsilon>0$. In dimension 3, the bound…

In our Novi Sad conference paper (1999) we described Dehn type surgeries of the famous Gieseking (1912) hyperbolic ideal simplex manifold $\mathcal{S}$, leading to compact fundamental domain $\mathcal{S}(k)$, $k = 2, 3, \dots$ with…

We investigate plane curves intersecting in at most two unibranched points to study the algebraic exceptional set appearing in standard conjectures of diophantine and hyperbolic geometry. Our first result compares the local geometry of two…

We prove that the minimal length of a closed geodesic with self-intersection number $k$ on any finite-type hyperbolic surface is $2\cosh^{-1}(1+2k)$ for $k>1750$. This improves the previously known threshold $k > 10^{13350}$. Our proof is…

Research about crossings is typically about minimization. In this paper, we consider \emph{maximizing} the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured…

In this paper, we prove the Bounded Height Conjecture which the author formulated in [2]. As a corollary, it follows that there are only a finite number of hyperbolic three manifolds of bounded volume and trace field degree.

We construct a small, hyperbolic 3-manifold $M$ such that, for any integer $g\geq 2$, there are infinitely many separating slopes $r$ in $\partial M$ so that $M(r)$, the 3-manifold obtained by attaching a 2-handle to $M$ along $r$, is…

By a work of Thurston, it is known that if a hyperbolic fibred $3$-manifold $M$ has Betti number greater than 1, then $M$ admits infinitely many distinct fibrations. For any fibration $\omega$ on a hyperbolic $3$-manifold $M$, the number of…

Associated to a hyperbolic knot complement in $S^3$ is a set of prime numbers corresponding to the residue characteristics of the ramified places of the quaternion algebras obtained by Dehn surgery on the knots. Previous work by…

We study the number and the length of systoles on complete finite area orientable hyperbolic surfaces. In particular, we prove upper bounds on the number of systoles that a surface can have (the so-called kissing number for hyperbolic…

We classify Dehn surgeries on (p,q,r) pretzel knots that result in a manifold of finite fundamental group. The only hyperbolic pretzel knots that admit non-trivial finite surgeries are (-2,3,7) and (-2,3,9). Agol and Lackenby's 6-theorem…

We show that any exceptional non-trivial Dehn surgery on a twist knot, except the trefoil, yields a 3-manifold whose fundamental group is left-orderable. This is a generalization of a result of Clay, Lidman and Watson, and also gives a new…

In this paper we study the difference between algebraic and geometric solutions of the hyperbolic Dehn filling equations for ideally triangulated 3-manifolds. We show that any geometric solution is an algebraic one, and we prove the…

We study thresholds for the appearance of a 2-core in random hypergraphs that are a mixture of a constant number of random uniform hypergraphs each with a linear number of edges but with different edge sizes. For the case of two overlapping…