arXiv subjects
Publications and source records attributed to Eaman Eftekhary.
We construct a new family of skein exact triangles for link Floer homology. The skein triples are described by a triple of rational tangles $(R_0,R_1,R_{2n+1})$, where $R_0$ is the trivial tangle and $R_k$ is obtained from it by applying $k$ positive half-twists. We also set up an appropriate framework for potential construction of further skein exact triangles corresponding to arbitrary triples of rational tangles.
We show that the torsion order $\mathrm{Ord}(K)$ of a knot $K$ in knot Floer homology gives a lower bound on the minimum number $n$ such that an oriented $(n+1)$-tangle replacement unknots $K$. This generalizes earlier results by Alishahi and the author and by Juhasz, Miller and Zemke, that $\mathrm{Ord}(K)$ is a lower bound for both the unknotting number $u(K)$ and for $br(K)-1$, where $br(K)$ denotes the bridge index of $K$.
We introduce a Heegaard-Floer homology functor from the category of oriented links in closed $3$-manifolds and oriented surface cobordisms in $4$-manifolds connecting them to the category of $\mathbb{F}[v]$-modules and $\mathbb{F}[v]$-homomorphisms between them, where $\mathbb{F}$ is the field with two elements. In comparison with previously defined TQFTs for decorated links and link cobordisms, the construction of this paper has the advantage of being independent from the decoration. Some of the basic properties of this functor are also explored.
The first author introduced a notion of equivalence on a family of $3$-manifolds with boundary, called (simple) balanced $3$-manifolds in an earlier paper and discussed the analogy between the Andrews-Curtis equivalence for group presentations and the aforementioned notion of equivalence. Motivated by the Andrews-Curtis conjecture, we use tools from Heegaard Floer theory to prove that there are simple balanced $3$-manifolds which are not in the trivial equivalence class (i.e. the equivalence class of $S^2\times [-1,1]$).
From the link Floer complex of a link $K$, we extract a lower bound $t_q'(K)$ for the rational unknotting number of $K$ (i.e. the minimum number of rational replacements required to unknot $K$). Moreover, we show that the torsion obstruction $t_q(K)=\hat{t}(K)$ from an earlier paper of Alishahi and the author is a lower bound for the proper rational unknotting number. Moreover, $t_q(K\#K')=\max\{t_q(K),t_q(K')\}$ and $t'_q(K\#K')=\max\{t'_q(K),t'_q(K')\}$. For the torus knot $K=T_{p,pk+1}$ we compute $t'_q(K)=\lfloor p/2\rfloor$ and $t_q(K)=p-1$.
We address the problem of counting periodic orbits of vector fields on smooth closed manifolds. The space of non-constant periodic orbits is enlarged to a complete space by adding the ghost orbits, which are decorations of the zeros of vector fields. Associated with any compact and open subset $\Gamma$ of the moduli space of periodic and ghost orbits, we define an integer weight. When the vector field moves along a path, and $\Gamma$ deforms in a compact and open family, we show that the weight function stays constant. We also give a number of examples and computations, which illustrate the applications of our main theorem.
Fix a smooth closed manifold $M$. Let $R_M$ denote the space of all pairs $(g,L)$ such that $g$ is a $C^3$ Riemannian metric on $M$ and the real number $L$ is not the length of any closed $g$-geodesics. A locally constant geodesic count function $\pi_M:R_M\rightarrow Z$ is constructed. For this purpose, the weight of compact open subsets of the space of closed $g$-geodesics is defined and investigated for an arbitrary Riemannian metric $g$.
In this short note, we compare the combinatorial sign assignment of Manolescu, Ozsvath, Szabo and Thurston for grid homology of knots and links in 3-sphere with the sign assignment coming from a coherent system of orientations on Whitney disks. Although these constructions produce different signs, a small modification of the convention in either of the two methods results in identical sign assignments, and thus identical chain complexes.
Given a knot K in S^3, let u^-(K) (respectively, u^+(K)) denote the minimum number of negative (respectively, positive) crossing changes among all unknotting sequences for K. We use knot Floer homology to construct the invariants l^-(K), l^+(K) and l(K), which give lower bounds on u^-(K), u^+(K) and the unknotting number u(K), respectively. The invariant l(K) only vanishes for the unknot, and is greater than or equal to the \nu^-(K). Moreover, the difference l(K)-\nu^-(K) can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number and the Gordian distance.
Let $K$ denote a knot inside the homology sphere $Y$ and $K'$ denote a knot inside a homology sphere $L$-space. Let $X=Y(K,K')$ denote the 3-manifold obtained by splicing the complements of $K$ and $K'$. We show that $\text{rank}(\widehat{HF}(X)) \ge \text{rank}(\widehat{HF}(Y))$.
We apply gauge theory to study the space $F_k(M)$ of smooth codimension-$k$ framed foliations on a smooth manifold $M$. The quotient of Maurer-Cartan elements by the action of an infinite dimensional non-abelian gauge groupoid forms a moduli space, which contains $F_k(M)$ as a subspace. The notion of holonomy is naturally extended to this moduli space and the cohomology theory associated with points of this moduli space which correspond to non-singular foliations coincides with Bott cohomology. The quotient of the moduli space under concordance is identified as the space of homotopy classes of maps to the classifying spaces $B\Gamma^g_k$ and $B\Gamma^q_k$. While $B\Gamma^g$ is a classic and has been studied since Haefliger, $B\Gamma^q$ (which is a quotient of $B\Gamma^g$) carries a simpler topology and offers a rival theory.
This note corrects the mistakes in the splicing formulas of the paper "Floer homology and splicing knot complements". The mistakes are the result of the incorrect assumption that for a knot $K$ inside a homology sphere $Y$, the involution on the knot Floer homology of $K$ which corresponds to moving the basepoints by one full twist around $K$ is trivial. The correction implicitly involves considering the contribution from this (possibly non-trivial) involution in a number of places.
We introduce a generalization of oriented tangles, which are still called tangles, so that they are in one-to-one correspondence with the sutured manifolds. We define cobordisms between sutured manifolds (tangles) by generalizing cobordisms between oriented tangles. For every commutative algebra A over Z/2Z, we define A-Tangles to be the category consisting of A-tangles, which are balanced tangles with A-colorings of the tangle strands and fixed SpinC structures, and A-cobordisms as morphisms. An A-cobordism is a cobordism with a compatible A-coloring and an affine set of SpinC structures. Associated with every A-module M we construct a functor $HF^M$ from A-Tangles to A-Modules, called the tangle Floer homology functor, where A-Modules denotes the the category of A-modules and A-homomorphisms between them. Moreover, for any A-tangle T the A-module $HF^M(T)$ is the extension of sutured Floer homology defined in an earlier work of the authors. In particular, this construction generalizes the 4-manifold invariants of Ozsvath and Szabo. Moreover, applying the above machinery to decorated cobordisms between links, we get functorial maps on link Floer homology.
Let $K$ denote a knot inside the homology sphere $Y$. The zero-framed longitude of $K$ gives the complement of $K$ in $Y$ the structure of a bordered three-manifold, which may be denoted by $Y(K)$. We compute the quasi-isomorphism type of the bordered Floer complex of $Y(K)$ in terms of the knot Floer complex associated with $K$. As a corollary, we show that if a homology sphere has the same Heegaard Floer homology as $S^3$ it does not contain any incompressible tori. Consequently, if $Y$ is an irreducible homology sphere $L$-space then $Y$ is either $S^3$, or the Poicar\'e sphere $\Sigma(2,3,5)$, or it is hyperbolic.
Let $\kappa_e(\overline{M}_{g,n})$ denote the kappa ring of $\overline{M}_{g,n}$ in codimension $e$. For $g,e\geq 0$ fixed, as the number $n$ of the markings grows large we show that the rank of $\kappa_e(\overline{M}_{g,n})$ is asymptotic to $$\frac{{n+e\choose e}{g+e\choose e}}{(e+1)!}\simeq \frac{{g+e\choose e}n^e}{e!(e+1)!}.$$ When $g\leq 2$ we show that a kappa class $\kappa$ is trivial if and only if the integral of $\kappa$ against all boundary strata is trivial. For $g=1$ we further show that the rank of $\kappa_{n-d}(\overline{M}_{1,n})$ is equal to $|P_1(d,n-d)|$, where $P_i(d,k)$ denotes the set of partitions $p=(p_1,...,p_\ell)$ of $d$ such that at most $k$ of the numbers $p_1,...,p_\ell$ are greater than $i$.
We obtain lower bounds on the rank of the kappa ring of the Delign-Mumford compactification of the moduli space of curves in different degrees. For this purpose, we introduce a quotient of the kappa ring, the combinatorial kappa ring, and show that the rank of this latter ring in degree $d$ is bounded below by $|P(d,3g-2+n-d)|$ where $P(d,r)$ denotes the set of partitions of the positive integer $d$ into at most $r$ parts. In codimension 1 (i.e. $d=3g-4+n$) we show that the rank of the kappa ring is equal to $n-1$ for $g=1$, and is equal to $\lceil \frac{(n+1)(g+1)}{2}\rceil-1$ for $g>1$. Furthermore, in codimension $e=3g-3+n-d$, the rank of the kappa ring (as $g$ and $e$ remain fixed and $n$ grows large) is asymptotic to $\frac{{n+e\choose e}{g+e\choose e}}{(e+1)!}$.
We introduce a refinement of the Ozsvath-Szabo complex associated to a balanced sutured manifold $(X,\tau)$ by Juhasz. An algebra $A_\tau$ is associated to the boundary of a sutured manifold and a filtration of its generators by $H^2(X,\partial X;\Z)$ is defined. For a fixed Spin^c structure $s$ over the manifold $X'$, which is obtained from $X$ by filling out the sutures, the Ozsvath-Szabo chain complex $CF(X,\tau,s)$ is then defined as a chain complex with coefficients in $A_\tau$ and filtered by $\SpinC(X,\tau)$. The filtered chain homotopy type of this chain complex is an invariant of $(X,\tau)$ and the Spin^c class $s\in\SpinC(X')$. The construction generalizes the construction of Juhasz. It plays the role of $CF^-(X,s)$ when $X$ is a closed three-manifold, and the role of $CFK^-(Y,K;s)$ when the sutured manifold is obtained from a knot $K$ inside a three-manifold $Y$. Our invariants generalize both the knot invariants of Ozsvath-Szabo and Rasmussen and the link invariants of Ozsvath and Szabo. We study some of the basic properties of the corresponding Ozsvath-Szabo complex, including the exact triangles, and some form of stabilization.
We show that if a positive integral surgery on a knot K inside a homology sphere X with Seifert genus g(K) results in an induced knot K_n in X_n(K)=Y which has simple Floer homology, we should have n>=2g(K). Moreover, if X is the standard sphere, the three-manifold Y is a L-space and the Heegaard Floer homology groups of K are determined by its Alexander polynomial.