arXiv research

Decomposes string links in a surface into prime components.

Study grid homology of diagonal knots, finding key terms related to prime factors and decompositions.

Decomposing knots and links into tangles is a useful technique for understanding their properties. The notion of prime tangles was introduced by Kirby and Lickorish in [3]; Lickorish proved [5] that by summing prime tangles one obtains a prime link. In a similar spirit, summing two prime alternating tangles will produc…

In this paper we use 3-manifold techniques to illuminate the structure of the string link monoid. In particular, we give a prime decomposition theorem for string links on two components as well as give necessary conditions for string links to commute under the stacking operation.

Study cohomotopy sets of simply connected 7-manifolds using suspension decompositions.

Study the moduli space of reducible 3-manifolds using prime decomposition.

Simplified proof classifies surfaces using normal curves.

This work classifies belted sum decompositions of fully augmented links.

This paper solves the structure of link concordance groups, proving they are infinitely generated.

We present an enhanced prime decomposition theorem for knots that gives the isotopy classes of composite knots that can be constructed from a given list of prime factors (allowing for the mirroring and orientation reversing for each factor). Underlying the theorem is an algebraic construction that also allows for the c…

Let k be a knot in S3. In [8], H.N. Howards and J. Schultens introduced a method to construct a manifold decomposition of double branched cover of (S3, k) from a thin position of k. In this article, we will prove that if a thin position of k induces a thin decomposition of double branched cover of (S3,k) by Howards and…

We study 2-string free tangle decompositions of knots with tunnel number two. As an application, we construct infinitely many counter-examples to a conjecture in the literature stating that the tunnel number of the connected sum of prime knots doesn't degenerate by more than one.

An open book decomposition of a 3-manifold $M$ induces a Heegaard splitting for $M$, and the minimal genus among all Heegaard splittings induced by open book decompositions is called the \emph{open book genus} of $M$. It is conjectured by Ozbagci \cite{O} that the open book genus is additive under the connected sum of …

We compute the Minimal Entropy of every closed, orientable $3$-manifold, showing that its cube equals the sum of the cubes of the minimal entropies of each hyperbolic component arising from the $JSJ$ decomposition of each prime summand. As a consequence we show that the cube of the Minimal Entropy is additive with resp…

We establish an existence and uniqueness theorem for prime decompositions of theta-curves in $3$-manifolds.

It has been shown by V. Colin that every tight contact 3-manifold can be written as a connected sum of prime manifolds. Here we prove that the summands in this decomposition are unique up to order and contactomorphism.

Unified framework for Alexandrov 3-spaces, extending manifold results.

Paper shows unique decomposition of 3-manifolds and multiplicative property of Reidemeister torsion.

We decompose into irreducible factors the ${\rm SU}(2)$ Witten-Reshetikhin-Turaev representations of the mapping class group of a genus $2$ surface when the level is $p=4r$ and $p=2r^2$ with $r$ an odd prime and when $p=2r_1r_2$ with $r_1$, $r_2$ two distinct odd primes. Some partial generalizations in higher genus are…

We show that a if a Riemannian manifold admits a universal cover with bounded geometry and if 0 does not belong to the spectrum or is an isolated point in the spectrum of the Laplacian on $\ell$-forms, then there exists $1<p<2$ such that for all $p<r<p^{\prime}$ the Hodge - de Rham decomposition for $L^{r}$-forms holds…

We enumerate a necessary condition for the existence of infinitely many geometrically distinct, non-constant, prime closed geodesics on an arbitrary closed Riemannian manifold $M$. That is, we show that any Riemannian metric on $M$ admits infinitely many prime closed geodesics such that the energy functional $E:ΛM\to\m…

We introduce mod 3 triple Milnor invariants and triple cubic residue symbols for certain primes of the Eisenstein number field $\mathbb{Q}(\sqrt{-3})$, following the analogies between knots and primes. Our triple symbol generalizes both the cubic residue symbol and Rédei's triple symbol, and describes the decomposition…

Let $M$ be a closed, oriented, connected 3--manifold and $(B,π)$ an open book decomposition on $M$ with page $Σ$ and monodromy $\varphi$. It is easy to see that the first Betti number of $Σ$ is bounded below by the number of $S^2\times S^1$--factors in the prime factorization of $M$. Our main result is that equality is…

We prove that for every integer k>1 there is a simply connected rational homology 5-sphere $\scriptstyle{M^5_k}$ with spin such that $\scriptstyle{H_2(M^5_k,\bbz)}$ has order $\scriptstyle{k^2},$ and $\scriptstyle{M^5_k}$ admits a Riemannian metric of positive Ricci curvature. Moreover, if the prime number decompositio…

We study a certain class of embedded two-foams that arise from gluing discs into ribbon torus knots along nonintersecting torus meridians. We exhibit several equivalent diagrammatic formalisms for these objects and identify several of their invariants, including a unique prime decomposition.

The chromatic number of sphere graphs in 3-manifolds is bounded.

Let G be the fundamental group of a connected, closed, orientable 3-manifold. We explicitly compute its virtually cyclic geometric dimension. Among the tools we use are the prime and JSJ decompositions of M, several push-out type constructions, as well as some Bredon cohomology computations.

The crushing operation of Jaco and Rubinstein is a powerful technique in algorithmic 3-manifold topology: it enabled the first practical implementations of 3-sphere recognition and prime decomposition of orientable manifolds, and it plays a prominent role in state-of-the-art algorithms for unknot recognition and testin…

Let $M$ be a closed, oriented and smooth manifold of dimension $d$. Let $ŁM$ be the space of smooth loops in $M$. Chas and Sullivan introduced loop product, a product of degree $-d$ on the homology of $LM$. In this paper we show how for three manifolds the ``nontriviality'' of the loop product relates to the ``hyperbol…

It is well-known that the monoid of long virtual knots is not commutative. This contrasts with the case of classical long knots, where $A \# B \leftrightharpoons B \# A$ for all $A,B$. In the present paper, we present a new proof that two inequivalent non-classical prime long virtual knots never commute. The original r…

Let $Σ_0,Σ_1$ be closed oriented surfaces. Two oriented knots $K_0 \subset Σ_0 \times [0,1]$ and $K_1 \subset Σ_1 \times [0,1]$ are said to be (virtually) concordant if there is a compact oriented $3$-manifold $W$ and a smoothly and properly embedded annulus $A$ in $W \times [0,1]$ such that $\partial W=Σ_1 \sqcup -Σ_0…

This paper classifies regular maps with Euler characteristic -p^4 for a prime p≥5.

This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces,…

We study the asymptotics of the number N(t) of geometrically distinct closed geodesics of a Riemannian or Finsler metric on a connected sum of two compact manifolds of dimension at least three with non-trivial fundamental groups and apply this result to the prime decomposition of a three-manifold. In particular we show…

We find decomposition series of length at most two for modular representations in positive characteristic of mapping class groups of surfaces induced by an integral version of the Witten-Reshetikhin-Turaev SO(3)-TQFT at the p-th root of unity, where p is an odd prime. The dimensions of the irreducible factors are given…

We construct instanton Floer homology for lens spaces $L(p,q)$. As an application, we prove that $X = \CP^2 # \CP^2$ does not admit a decomposition $X = X_1 \cup X_2$. Here $X_1$ and $X_2$ are oriented, simply connected, non-spin 4-manifolds with $b^+ = 1$ and with boundary $L(p, 2)$, and $p$ is a prime number of the f…

Let G be a graph in a 3-manifold M. We compress the pair (M,G) along admissible 2-spheres as long as possible. What we get is a root of (M,G). Our main result is that for any pair (M,G) the root exists and is unique. As a corollary we get an easy proof of Petronio's theorem on prime decompositions of 3-orbifolds.

The paper proves a free product decomposition for congruence subgroups with constraints.

Study primes dividing torsion in homology of commuting elements in Lie groups.

The cohomology ring with coefficients in $\Z_p$, where $p$ is a prime integer, of a Seifert manifold $M$, orientable or not orientable is obtained from a simplicial decomposition of $M$. Many choices must be made before applying Alexander-Whitney formula to get the cup-products. The most difficult choices are those of …

Study on singularities of area-minimizing currents, focusing on frequency and branch points.

For all n > 0 there is a homomorphism from the smooth concordance group of knots in dimension 2n + 1 to an algebraically defined group called the rational algebraic concordance group. This algebraic concordance group splits as a direct sum of groups indexed by polynomials. For n > 1 the homomorphism is injective. This …

Study volume conjecture for links with multiple hyperbolic pieces.

Let M be a closed oriented three-manifold, whose prime decomposition contains no aspherical factors. We show that for any initial riemannian metric on M the solution to the Ricci flow with surgery, defined in our previous paper math.DG/0303109, becomes extinct in finite time. The proof uses a version of the minimal dis…

Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.

Study open 3-manifolds as sums of closed ones, finding a classification.

Proves prime theta-curves for knots on minimal genus surfaces.

It is a major unsolved problem as to whether unknot recognition - that is, testing whether a given closed loop in R^3 can be untangled to form a plain circle - has a polynomial time algorithm. In practice, trivial knots (which can be untangled) are typically easy to identify using fast simplification techniques, wherea…