Narrow Results

We generalize Kauffman’s famous formula defining the Jones polynomial of an oriented link in $3$-space from his bracket and the writhe of an oriented diagram [L. Kauffman, State models and the Jones polynomial, Topology26(3) (1987) 395–407]. Our generalization is an epimorphism between skein modules of tangles in compact connected oriented $3$-manifolds with markings in the boundary. Besides the usual Jones polynomial of oriented tangles we will consider graded quotients of the bracket skein module and Przytycki’s $q$-analog of the first homology group of a $3$-manifold [J. Przytycki, A $q$-analogue of the first homology group of a $3$-manifold, in Contemporary Mathematics, Vol. 214 (American Mathematical Society, 1998), pp. 135–144]. In certain cases, e.g., for links in submanifolds of rational homology $3$-spheres, we will be able to define an epimorphism from the Jones module onto the Kauffman bracket module. For the general case we define suitably graded quotients of the bracket module, which are graded by homology. The kernels define new skein modules measuring the difference between Jones and bracket skein modules. We also discuss gluing in this setting.

In this paper, we aim to interpret the background gravitational effects appearing in quantum field theory on curved space-time by studying the Brownian motion of quantum states along with the Hamilton–Perelman Ricci flow. It has been shown that the Wiener measure automatically contains the Einstein–Hilbert action and the path-integral formulation of the scalar quantum field theory on curved space-time at the first order of local approximations. This provides a well-defined formulation of the path-integral measure for quantum field theory in the presence of gravity. However, we establish that the emergence of Einstein–Hilbert action is independent of the matter field interactions and is a merely entropic/geometric effect stemming from the nature of the Ricci flow of the universe geometry. We also extract an explicit formula for the cosmological constant in terms of the Ricci flow and Hamilton’s theorem for 3-manifolds. Then, we discuss the cosmological features of the FLRW solution in $\mathrm{\Lambda }$CDM Model via the derived equations of the Ricci flow. We also argue the correlation between our formulations and the entropic aspects of gravity. Finally, we provide some theoretical evidence that proves the second law of thermodynamics is the basic source of gravity and probably a more fundamental concept.

In this paper, we prove that for any closed, connected, oriented $3$-manifold $M$, there exists an infinite family of 2-fold branched covers of $M$ that are hyperbolic 3-manifolds and surface bundles over the circle with arbitrarily large volume.

The paper relates the 4-fold symmetric quandle homotopy (cocycle) invariants to topological objects. We show that the 4-fold symmetric quandle homotopy invariants are at least as powerful as the Dijkgraaf–Witten invariants. As an application, for an odd prime p, we show that the quandle cocycle invariant of a link in S³ constructed by the Mochizuki 3-cocycle is equivalent to the Dijkgraaf–Witten invariant with respect to ℤ/pℤ of the double covering of S³ branched along the link. We also reconstruct the Chern–Simons invariant of closed 3-manifolds as a quandle cocycle invariant via the extended Bloch group, in analogy to [A. Inoue and Y. Kabaya, Quandle homology and complex volume, preprint(2010), arXiv:math/1012.2923].

The intersecting kernel of a Heegaard splitting $H_1{\bigcup }_S{H}_2$ for a compact orientable 3-manifold $M$ is the subgroup $K=\mathrm{\text{Ker}}({i_1}_{\ast })\cap \mathrm{\text{Ker}}({i_2}_{\ast })$ of ${\pi }_1(S)$, where ${i_j}_{\ast }:{\pi }_1(S)\to {\pi }_1(H_j)$ is the homomorphism induced by the inclusion $i_j:S\hookrightarrow H_j$, $j=1,2$. In the paper, we obtain some invariants of 3-manifolds $M$ from certain quotient groups of the intersecting kernels of their Heegaard splittings. We also list two algebraic problems related to the new invariants, which might be interesting to study.

In the setting of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries, there are two candidates to be universal invariants, defined, respectively, by Kricker and Lescop. In a previous paper, the second author defined maps between spaces of Jacobi diagrams. Injectivity for these maps would imply that Kricker and Lescop invariants are indeed universal invariants; this would prove in particular that these two invariants are equivalent. In the present paper, we investigate the injectivity status of these maps for degree 2 invariants, in the case of knots whose Blanchfield modules are direct sums of isomorphic Blanchfield modules of $\mathbb{Q}$-dimension two. We prove that they are always injective except in one case, for which we determine explicitly the kernel.

Suppose that $\Sigma$ is a closed and oriented surface equipped with a Riemannian metric. In the literature, there are three seemingly distinct constructions of open books on the unit (co)tangent bundle of $\Sigma$, having complex, contact and dynamical flavors, respectively. Each one of these constructions is based on either an admissible divide or an ordered Morse function on $\Sigma$. We show that the resulting open books are pairwise isotopic provided that the ordered Morse function is adapted to the admissible divide on $\Sigma$. Moreover, we observe that if $\Sigma$ has positive genus, then none of these open books are planar and furthermore, we determine the only cases when they have genus one pages.

It is known that the LMO invariant of 3-manifolds with positive first Betti numbers is relatively weak and can be determined by “(semi-)classical” invariants such as the cohomology ring, the Alexander polynomial, and the Casson–Walker–Lescop invariant.

In this paper, we formulate a refinement of the LMO invariant for 3-manifolds with the first Betti number 1. It dominates the perturbative SO(3) invariant of such 3-manifolds, which is the power series invariant formulated by the arithmetic perturbative expansion of the quantum SO(3) invariants of such 3-manifolds. As the 2-loop part of the refinement of the LMO invariant, we define the 2-loop polynomial of such 3-manifolds. Further, as the $𝔰{𝔩}_m$ reduction at large $m$ limit of the $\ell$-loop part of the refinement of the LMO invariant for $\ell \le 5$, we formulate an $\ell$-variable polynomial invariant of such 3-manifolds whose Alexander polynomial is constant.

Metric conditions "C_α" and "" are defined for finite group presentations. If the fundamental group of a closed aspherical 3-manifold has some presentation which satisfies C₂ or , then its universal cover is simply connected at infinity. These ideas are derived from work by A. Casson and V. Poénaru.

An uncountable collection of arcs in S³ is constructed, each member of which is wild precisely at its endpoints, such that the fundamental groups of their complements are non-trivial, pairwise non-isomorphic, and indecomposable with respect to free products. The fundamental group of the complement of a certain Fox-Artin arc is also shown to be indecomposable.

Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpreted as the skein module of the 3-manifold obtained by gluing the two handlebodies together along this surface. A spectral sequence associated to the Hochschild complex is constructed and conditions are given for the existence of algebraic torsion in the completion of the skein module of this 3-manifold.

A census is presented of all closed non-orientable 3-manifold triangulations formed from at most seven tetrahedra satisfying the additional constraints of minimality and ℙ²-irreducibility. The eight different 3-manifolds represented by these 41 different triangulations are identified and described in detail, with particular attention paid to the recurring combinatorial structures that are shared amongst the different triangulations. Using these recurring structures, the resulting triangulations are generalised to infinite families that allow similar triangulations of additional 3-manifolds to be formed.

In the ordinary normal surface for a compact 3-manifold, any incompressible, ∂-incompressible, compact surface can be moved by an isotopy to a normal surface [9]. But in a non-compact 3-manifold with an ideal triangulation, the existence of a normal surface representing an incompressible surface cannot be guaranteed. The figure-8 knot complement is presented in a counterexample in [12]. In this paper, we show the existence of normal Seifert surface under some restriction for a given ideal triangulation of the knot complement.

From a pseudo-triangulation with n tetrahedra T of an arbitrary closed orientable connected 3-manifold (for short, a 3D-space) M³, we present a gem J′, inducing 𝕊³, with the following characteristics: (a) its number of vertices is O(n); (b) it has a set of p pairwise disjoint couples of vertices {u_i, v_i}, each named a twistor; (c) in the dual (J′)^⋆ of J′ a twistor becomes a pair of tetrahedra with an opposite pair of edges in common, and it is named a hinge; (d) in any embedding of (J′)^⋆ ⊂ 𝕊³, the ∊-neighborhood of each hinge is a solid torus; (e) these p solid tori are pairwise disjoint; (f) each twistor contains the precise description on how to perform a specific surgery based in a Denh–Lickorish twist on the solid torus corresponding to it; (g) performing all these p surgeries (at the level of the dual gems) we produce a gem G′ with |G′| = M³; (h) in G′ each such surgery is accomplished by the interchange of a pair of neighbors in each pair of vertices: in particular, |V(G′) = |V(J′)|.

This is a new proof, based on a linear polynomial algorithm, of the classical theorem of Wallace (1960) and Lickorish (1962) that every 3D-space has a framed link presentation in 𝕊³ and opens the way for an algorithmic method to actually obtaining the link by an O(n²)-algorithm. Actually this has been done and awaits a proper implentation.

This paper gives a proof that the universal cover of a closed 3-manifold built from three π₁-injective handlebodies is homeomorphic to ℝ³. This construction is an extension to handlebodies of the conditions for gluing of three solid tori to produce non-Haken Seifert fibered manifolds with infinite fundamental group. This class of manifolds has been shown to contain non-Haken non-Seifert fibered manifolds.

Gauss diagram formulas are extensively used to study Vassiliev link invariants. Now we apply this approach to invariants of 3-manifolds, considering manifolds given by surgery on framed links in the 3-sphere. We study the lowest degree case — the celebrated Casson–Walker invariant of rational homology spheres. This paper is dedicated to a detailed treatment of 2-component links; a general case will be considered in a forthcoming paper. We present simple Gauss diagram formulas for the Casson–Walker invariant. This enables us to understand/separate its dependence on the unframed link and on the framings. We also obtain skein relations for the Casson–Walker invariant under crossing changes, and study its asymptotic behavior when framings tend to infinity. Finally, we present results of extensive computer calculations.

We introduce a complexity c(X) ∈ ℕ for pairs X = (M,L), where M is a closed orientable 3-manifold and L ⊂ M is a link. The definition employs simple spines, but for well-behaved X's, we show that c(X) equals the minimal number of tetrahedra in a triangulation of M containing L in its 1-skeleton. Slightly adapting Matveev's recent theory of roots for graphs, we carefully analyze the behaviour of c under connected sum away from and along the link. We show in particular that c is almost always additive, describing in detail the circumstances under which it is not. To do so we introduce a certain (0, 2)-root for a pair X, we show that it is well-defined, and we prove that X has the same complexity as its (0, 2)-root. We then consider, for links in the 3-sphere, the relations of c with the crossing number and with the hyperbolic volume of the exterior, establishing various upper and lower bounds. We also specialize our analysis to certain infinite families of links, providing rather accurate asymptotic estimates.

We prove for the Reidemeister–Turaev torsion of closed oriented three-manifolds some finiteness properties in the sense of Goussarov and Habiro, that is, with respect to some cut-and-paste operations which preserve the homology type of the manifolds. In general, those properties require the manifolds to come equipped with an Euler structure and a homological parametrization.

We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils interesting properties: it is subadditive under connected sum and finite-to-one on ℙ²-irreducible manifolds. Moreover, for ℙ²-irreducible manifolds, it equals the minimal number of cubes in a cubulation of the manifold, except for the sphere S³, the projective space ℝℙ³ and the lens space L_4,1, which have surface-complexity zero. We will also give estimations of the surface-complexity by means of triangulations, Heegaard splittings, surgery presentations and Matveev complexity.

We define two kinds of invariants of links in closed 3-manifolds, the s-complexity(s ∈ ℕ) and the block number, by considering decompositions of links in closed orientable 3-manifolds by spines. The first one is a generalization of the complexity of links defined by Pervova and Petronio. After providing properties of these invariants, we construct special spines of strongly-cyclic coverings branched over generalized twist knots in lens spaces, including S³ and ℝP³, which provide upper bounds for the invariants.